Learning Middle School Mathematics Through Student Designed and Constructed Video Games

UNLV Theses, Dissertations, Professional Papers, and Capstones

5-2011

Learning middle school mathematics through student designed and constructed video games

Camille M. McCue University of Nevada, Las Vegas

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Repository Citation McCue, Camille M., "Learning middle school mathematics through student designed and constructed video games" (2011). UNLV Theses, Dissertations, Professional Papers, and Capstones. 919. http://dx.doi.org/10.34917/2261383

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This Dissertation has been accepted for inclusion in UNLV Theses, Dissertations, Professional Papers, and Capstones by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. LEARNING MIDDLE SCHOOL MATHEMATICS THROUGH STUDENT

DESIGNED AND CONSTRUCTED VIDEO GAMES

Camille Moody McCue

Bachelor of Arts University of Texas, Austin 1988

Master of Arts University of Texas, San Antonio 1993

A dissertation submitted in partial fulfillment of the requirements for the

Doctor of Philosophy in Curriculum & Instruction Department of Curriculum & Instruction College of Education

Graduate College University of Nevada, Las Vegas May 2011

THE GRADUATE COLLEGE

We recommend the dissertation prepared under our supervision by

Camille Moody McCue

entitled

Learning Middle School Mathematics Through Student Designed and Constructed Video Games

be accepted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Curriculum and Instruction

Randall Boone, Committee Chair

Kent Crippen, Committee Member

PG Schrader, Committee Member

David James, Graduate Faculty Representative

Ronald Smith, Ph. D., Vice President for Research and Graduate Studies and Dean of the Graduate College

May 2011

ii ABSTRACT

Learning Middle School Mathematics Through Student

Designed and Constructed Video Games

Camille Moody McCue

Dr. Randall Boone, Examination Committee Chair Professor of Curriculum and Instruction University of Nevada, Las Vegas

Mathematics achievement is an area in which American precollege students are faltering. Emerging research suggests that making mathematics instruction relevant and applicable in the lives of youth may impact math achievement, especially when it capitalizes on high-interest technologies such as video games.

Employing a quasi-experimental and descriptive approach, this study examined the mathematics (i.e., numbers and operations, algebra, geometry, measurement, and probability) that middle school students employed during their design and construction of video games. First, it examined the mathematics content learned by 19 sixth and seventh graders during their analysis, synthesis, and programming of three video game projects over 7 months. Second, it measured the ability of the student programmers to laterally transfer mathematics content from the technology context of game production to the traditional context of paper-and-pencil tests. Third, it evaluated student attitudes toward mathematics prior to and following video game design and construction. The performance of student programmers was compared with that of a control group of nonprogrammers on measures of transfer and affect.

iii Results indicated that middle grade students successfully identified the events defining game play (e.g., motion, collisions, and scoring) of three, simple video game models. They successfully represented video game events in both mathematical and programming forms by writing and coding (a) boundary conditions using inequalities, (b) coordinate locations and identification of coordinate convergence, (c) directional headings, (d) uniform linear motion, (e) variable changes, and (f) probability-based consequences. They were also successful in writing programming code for their own functional video games, with a high percentage of relevant mathematics content incorporated therein. However, while treatment students transferred mathematical knowledge from the technology to the traditional context, it appeared that, without explicit bridging, the transfer was no better than comparison students. Treatment students also demonstrated no significant changes in attitude associated with designing and constructing video games. This study demonstrated that video game design and construction can be a viable – although not significantly different – method, cognitively and affectively, of instructing age-appropriate, standards-based mathematics content.

iv ACKNOWLEDGEMENTS

Thank you to my extraordinary family – Ian, Carson, and Michael – for their constant love, support, and understanding during this undertaking. Nothing really matters without you all, so I am grateful that you also believed that my efforts in pursuing a doctorate were a valuable investment in the future.

Thanks to my parents, Beverly Dempsey-Moody and Dr. Eric Moody, for serving as role models all my life. Thanks to my friends and colleagues who share the same hope and enthusiasm for transforming education into something our youth find worthy and fulfilling.

I would also like to express enormous gratitude to my students who enrolled in math enrichment and served as the treatment group. This study simply would not have been possible without your kindness and diligence in helping me execute my research.

And thanks to my outstanding dissertation committee, Dr. Randy Boone, Dr. Kent

Crippen, Dr. PG Schrader, and Dr. Dave James, for your advice, guidance, and encouragement over the many years it has taken to bring this to fruition.

Dedicated with love to my sons,

Carson McCue and Ian McCue.

vi TABLE OF CONTENTS

ABSTRACT...... iii

ACKNOWLEDGEMENTS...... v

DEDICATION...... vi

LIST OF TABLES...... xiv

LIST OF FIGURES ...... xvi

CHAPTER 1 INTRODUCTION ...... 1 Background...... 1 Video Games...... 1 Video Games and Mathematics ...... 3 Youth-Oriented Programming Environments...... 4 Selection of Programming Projects ...... 5 Attitudes Toward Mathematics...... 6 Conceptual Underpinnings for the Study...... 6 Recognizing Mathematics Inherent in the Design of Video Games...... 7 Learning Complex Tasks...... 9 Lateral Transfer of Mathematics...... 10 Attitudes Toward Math and the Role of Attitude in Achievement in Mathematics...... 12 Statement of the Problem...... 13

Purpose of the Study ...... 15 Research Questions...... 15 The mathematics of video game design and construction ...... 15

The transfer of mathematics ability ...... 16

Attitudes toward mathematics ...... 16

Hypotheses ...... 16 Hypothesis regarding the mathematics of video game

design and construction ...... 16

Hypothesis regarding the transfer of mathematics ability ...... 17

Hypothesis regarding attitudes toward mathematics ...... 17 Definitions ...... 17

CHAPTER 2 REVIEW OF RELATED LITERATURE...... 21 Introduction...... 21 Literature Addressing the Mathematics of Video Games...... 21 Rationale for the Video Game Context...... 22 Theories of Video Game Design and Construction ...... 23

vii Video game design: contributions of James Gee ...... 24 Video game construction: contributions of Jeroen van Merriënboer ...... 28 Programming Environment Options for Preteenagers...... 31 Focus on Math Instruction in the Video Game Construction Context ...... 36 Literature Relevant to the Transfer of Mathematics Ability...... 38 The Significance of Measuring Mathematical Achievement ...... 39 Mathematics Standards in the Middle Grades ...... 40 Mathematics content standards ...... 40 Role of the content standards in video game design and construction ...... 42 Mathematics process standards ...... 42 Role of the process standards in video game design and construction...... 43 Curricula and Instructional Methods in Mathematics...... 44 Lateral Transfer from a Related Subject Area to Mathematics ...... 47 Literature Relevant to Student Affect Associated with Mathematics...... 53 Definition of Affect ...... 54 Instruments for Measuring Affect...... 55 Historical affective instruments ...... 55 Criteria for distinguishing quality affective instruments ...... 56 Attitudes Toward Mathematics Inventory (ATMI) ...... 57 Affect and Achievement ...... 57

CHAPTER 3 METHODOLOGY ...... 60 Research Questions...... 61 Research Design ...... 62 Design Philosophy ...... 63 Overview of Quantitative Methods...... 64 Standardized tests in mathematics ...... 65 Studywide mathematics content tests (pre/post study)...... 65 Checkpoint mathematics content tests (pre/post video game project)...... 65 Event-recognition tallies ...... 65 Representations scores...... 66 Game model events included...... 66 Modifications included ...... 66 Attitude inventory scores (pre/post treatment) ...... 66 Overview of Qualitative Methods...... 66 Self-reflections (pre/post video game project)...... 67 Observations ...... 67 Talk-alouds ...... 67 Evaluation of completed video games ...... 68 Quantitative Methods Regarding Research Question 1a: Video Game Analysis ...... 68 Event recognition tallies ...... 68 Quantitative Methods Regarding Research Question 1b: Video Game Synthesis...... 69 Representations scores...... 69

viii Quantitative Methods Regarding Research Question 1c: Video Game Programming ...... 69 Qualitative Methods Regarding Research Question 1: Design and Construction...... 70 Self-reflections...... 70 Completed video game files ...... 71 Observations ...... 71 Talk-alouds ...... 72 Quantitative Methods Regarding Research Question 2: Transfer of Math Content ...... 72 Standardized tests ...... 72 Studywide mathematics content tests ...... 73 Checkpoint mathematics content tests...... 74 Quantitative and Qualitative Methods Regarding Research Question 3: Attitudes...... 74 Attitude scores ...... 75 Observations and talk-alouds...... 76 Study Participants ...... 76 Treatment Group...... 76 Comparison Group...... 76 Non-Random Assignment and Initial Differences...... 77 Setting ...... 77 Data Collection ...... 79 Sequence of Study Activities...... 79 Outset, data-collection ...... 80 Treatment period, general overview ...... 80 Treatment period, data collection, pre-analysis ...... 81 Treatment period, data collection, analysis ...... 82 Treatment period, data collection, synthesis...... 82 Treatment period, lessons learned from pilot study...... 83 Treatment period, data collection, pre-programming ...... 83 Treatment period, data collection, programming...... 84 Treatment period, data collection, post-programming...... 85 Close of study, data collection...... 85 Study Timeline...... 85 Instruments...... 86 Instruments, Studywide Mathematics Content Tests (Pre/Post Study) ...... 86 Instruments, Checkpoint Math Content Tests (Pre/Post Study) ...... 89 Instruments, Student Game Design Journals ...... 89 Instruments, Video Game Files ...... 90 Instruments, Attitudes Inventory ...... 91 Mitigating Potential Threats to Validity ...... 92 Threats to Validity, Role of the Researcher...... 92 Threats to Validity, Sample Selection ...... 94 Threats to Validity, Instruments ...... 94 Threats to Validity, Maturation and Morbidity ...... 94

ix Data Analysis of Quantitative Measures ...... 95 Data Analysis of Research Question 1: Event Tallies, Representations, Event Inclusions, Game Modifications ...... 96 Data Analysis of Research Question 2: Lateral Transfer of Math Ability ...... 98 Analysis of lateral transfer of math within the treatment group ...... 98 Analysis of lateral transfer of math between groups ...... 99 Data Analysis of Research Question 3: Attitudes Toward Mathematics...... 101

CHAPTER 4 RESULTS ...... 103 Summary of Research Questions...... 103 Question 1, Mathematics of Video Game Design and Construction...... 103 Question 2, Lateral Transfer of Mathematics Content Knowledge ...... 103 Question 3, Attitudes Toward Mathematics ...... 104 Initial Differences Between Groups ...... 104 Comparison by Achievement Test Scores ...... 105 Achievement test scores, descriptive statistics ...... 105 Achievement test scores, tests of normality...... 106 Achievement test scores, differences between groups...... 107 Comparison by Studywide Content Pre-test Scores ...... 107 Studywide content pre-test scores, descriptive statistics ...... 107 Studywide content pre-test scores, tests of normality...... 108 Studywide content pre-test scores, differences between groups...... 109 Comparison by ATMI Affective Scale Scores ...... 109 Pre-treatment ATMI scores, descriptive statistics ...... 110 Pre-treatment ATMI scores, tests of normality ...... 111 Pre-treatment ATMI scores, differences between groups ...... 114 Data Obtained from the Etch-a-Sketch Digital Toy ...... 114 Etch-a-Sketch Checkpoint Tests...... 115 Etch-a-Sketch Initial Events Tally...... 116 Etch-a-Sketch Representations ...... 119 Etch-a-Sketch representation ratings by type ...... 119 Etch-a-Sketch representation ratings by standard...... 120 Etch-a-Sketch Initial Reflections and Design Plans ...... 120 Statements regarding successes ...... 121 Statements regarding challenges...... 121 Statements regarding planned modifications...... 121 Etch-a-Sketch Digital Toy Programming ...... 123 Model events included...... 125 Modifications included ...... 126 Etch-a-Sketch Final Reflections ...... 127 Statements regarding successes ...... 127 Statements regarding challenges...... 128 Statements regarding implemented modifications...... 129 Data Obtained from the Frogger Video Game...... 130

x Frogger Checkpoint Tests...... 130 Frogger Initial Events Tally...... 131 Frogger Representations ...... 133 Frogger representation ratings by type ...... 135 Frogger representation ratings by standard...... 136 Frogger Initial Reflections and Design Plans ...... 137 Statements regarding successes ...... 137 Statements regarding challenges...... 138 Statements regarding modifications...... 139 Frogger Video Game Programming ...... 140 Game model events included...... 140 Modifications included ...... 142 Frogger Final Reflections ...... 142 Statements regarding successes ...... 142 Statements regarding challenges...... 143 Statements regarding modifications...... 143 Data Obtained from Tamagotchi Virtual Pet Video Game...... 145 Tamagotchi Checkpoint Tests ...... 145 Tamagotchi Initial Events Tally ...... 147 Tamagotchi Representations...... 149 Tamagotchi representation ratings by type...... 151 Tamagotchi representation ratings by standard ...... 152 Tamagotchi Initial Reflections and Design Plans...... 153 Statements regarding successes ...... 153 Statements regarding challenges...... 153 Statements regarding modifications...... 154 Tamagotchi Video Game Programming ...... 157 Game model events included...... 158 Modifications included ...... 159 Tamagotchi Final Reflections...... 160 Statements regarding successes ...... 160 Statements regarding challenges...... 161 Statements regarding modifications...... 161 Studywide Post-test Performance ...... 167 Studywide Questions Derived from Etch-a-Sketch ...... 168 Treatment group performance ...... 168 Comparison group performance ...... 169 Treatment vs. comparison group performance ...... 170 Studywide Questions Derived from Frogger ...... 171 Treatment group performance ...... 171 Comparison group performance ...... 172 Treatment vs. comparison group performance ...... 173 Studywide Questions Derived from Tamagotchi...... 174 Treatment group performance ...... 174 Comparison group performance ...... 175 Treatment vs. comparison group performance ...... 176

xi Studywide Content Tests, Pre-test-to-Post-test Changes...... 177 Treatment group performance ...... 177 Comparison group performance ...... 178 Treatment vs. comparison group performance ...... 179 Post-treatment ATMI Inventory Outcomes ...... 181 Studywide Pre-treatment-to-Post-treatment ATMI Scores (Within Groups) ...... 182 Studywide Pre-treatment-to-Post-treatment ATMI Scores (Between Groups) .... 184 Other Correlational Relationships ...... 185

CHAPTER 5 DISCUSSION...... 186 Summary of Research Questions...... 186 Question 1, Mathematics of Video Game Design and Construction...... 186 Question 2, Lateral Transfer of Mathematics Content Knowledge ...... 186 Question 3, Attitudes Toward Mathematics ...... 187 Question 1a – Mathematics of Video Game Design and Construction: Analysis ...... 187 Analyzing Video Game Events...... 188 Tallies of Initial Events by Project ...... 188 Question 1b – Mathematics of Video Game Design and Construction: Synthesis ...... 189 Synthesizing Representations in Etch-a-Sketch ...... 189 Synthesizing Representations in Frogger ...... 190 Synthesizing Representations in Tamagotchi Virtual Pet...... 191 Question 1c – Mathematics of Video Game Design and Construction: Programming ...... 192 Programming Etch-a-Sketch...... 193 Etch-a-Sketch model events...... 193 Etch-a-Sketch modifications...... 194 Programming Frogger...... 194 Frogger model events...... 194 Frogger modifications...... 196 Programming Tamagotchi Virtual Pet...... 196 Tamagotchi model events ...... 196 Tamagotchi modifications ...... 197 Support of the Hypothesis Regarding Video Game Design and Construction...... 198 Question 2 – Lateral Transfer of Mathematics Content Knowledge ...... 198 Question 2a – Lateral Transfer of Mathematics, Within Treatment Group...... 199 Lateral transfer by content standard (treatment group) ...... 199 Lateral transfer by project (treatment group)...... 200 Question 2b – Lateral Transfer of Mathematics, Between Groups Comparison ...... 201 Studywide shifts by raw score ...... 201 Studywide shifts by mathematics content standards...... 202 Support of the Hypothesis Regarding Transfer of Math Content Knowledge ...... 202 Question 3 – Attitudes Toward Mathematics ...... 204

xii Question 3a – Attitudes Within the Treatment Group...... 204 Question 3b – Attitudes Between Groups...... 205 Rejection of the Hypothesis Regarding Attitudes Toward Mathematics ...... 206 Conclusions...... 206 Limitations...... 209 Recommendations for Future Research...... 212

APPENDICES ...... 214 A. EXAMPLE TREATMENT STUDENT PROFILES...... 214 B. STUDYWIDE MATHEMATICS CONTENT TEST ...... 218 C. ATTITUDES TOWARD MATHEMATICS INVENTORY (ATMI) ...... 226 D. CHECKPOINT MATHEMATICS TESTS...... 229 E. EVENTS LIST TEMPLATE...... 240 F. EXAMPLE REPRESENTATIONS TEMPLATE (ETCH-A-SKETCH) ...... 242 G. REFLECTIONS AND PLANS DESIGN JOURNAL TEMPLATES ...... 245 H. TRANSCRIPT OF CLASS DISCUSSION (ETCH-A-SKETCH)...... 248 I. TAMAGOTCHI VIDEO GAME MODEL PROGRAMMING CODE WITH GAPS ...... 258 J. UNLV IRB APPROVALS ...... 260 K. CONSENT FORMS ...... 264

REFERENCES ...... 271

VITA...... 290

xiii LIST OF TABLES

Table 1 Example Video Game Events and Associated NCTM Content Standards...... 19 Table 2 Example Event Representations in Tamagotchi Virtual Pet...... 19 Table 3 Mathematics Achievement Test Scores by Group...... 105 Table 4 Tests of Normality: Mathematics Achievement Test Scores by Group...... 106 Table 5 Studywide Mathematics Content Pre-test Scores by Group...... 107 Table 6 Tests of Normality: Studywide Mathematics Content Pre-test Scores by Group...... 108 Table 7 ATMI Scales and Maximum Possible Values ...... 110 Table 8 Pre-treatment ATMI Inventory Scores by Scale and by Group...... 111 Table 9 Tests of Normality: ATMI Pre-treatment Scores Scores by Scale and by Group ...... 113 Table 10 Etch-a-Sketch Checkpoint Test Performance, Pre-test to Post-test...... 116 Table 11 Tally of Events Recorded for Etch-a-Sketch Digital Toy Model ...... 117 Table 12 Sample Representations Provided for the Etch-a-Sketch ...... 118 Table 13 Ratings of All Representations Recorded for Etch-a-Sketch...... 119 Table 14 Ratings of Mathematical Representations (by Standard) Recorded for the Etch-a-Sketch ...... 120 Table 15 Tally of Etch-a-Sketch Model Events Included in Video Games...... 125 Table 16 Etch-a-Sketch Modifications Constructed ...... 126 Table 17 Frogger Checkpoint Test Performance (Pre-test to Post-test) ...... 131 Table 18 Tally of Events Recorded for Frogger Video Game Model During Game Analysis...... 132 Table 19 Sample Representations Provided for the Frogger Video Game...... 135 Table 20 Representations (by Type) Recorded for Frogger ...... 136 Table 21 Mathematical Representations (by Standard) Recorded for Frogger ...... 137 Table 22 Events from Revised List that were Included in Frogger Video Games...... 141 Table 23 Frogger Modifications Constructed...... 142 Table 24 Tamagotchi Checkpoint Test Performance (Pre-test to Post-test)...... 146 Table 25 Tally of Events Recorded for the Tamagotchi Video Game Model During Game Analysis...... 148 Table 26 Sample Representations Provided for the Tamagotchi Video Game ...... 150 Table 27 Representations Recorded for Tamagotchi Virtual Pet ...... 151 Table 28 Mathematical Representations (by Type) Recorded for Tamagotchi Virtual Pet...... 152 Table 29 Events from Revised List that were Included in Tamagotchi Video Games ...... 159 Table 30 Tamagotchi Modifications Constructed...... 160 Table 31 Studywide Content Post-test Scores by Group...... 168 Table 32 Studywide Content, Etch-a-Sketch Subset (Pre-test to Post-test) – Treatment Group ...... 169

xiv Table 33 Studywide Content, Etch-a-Sketch Subset (Pre-test to Post-test) – Comparison Group...... 170 Table 34 Studywide Content, Etch-a-Sketch Subset (Pre-test to Post-test) – All Participants...... 171 Table 35 Studywide Content, Frogger Subset (Pre-test to Post-test) – Treatment Group ...... 172 Table 36 Studywide Content, Frogger Subset (Pre-test to Post-test) – Comparison Group...... 173 Table 37 Studywide Content, Frogger Subset (Pre-test to Post-test) – All Participants...... 173 Table 38 Studywide Content, Tamagotchi Subset (Pre-test to Post-test) – Treatment Group ...... 175 Table 39 Studywide Content, Tamagotchi Subset (Pre-test to Post-test) – Comparison ...... 175 Table 40 Studywide Content, Tamagotchi Subset (Pre-test to Post-test) – All participants...... 176 Table 41 Studywide Content Test (Pre-test to Post-test) – Treatment Group ...... 177 Table 42 Studywide Content Test (Pre-test to Post-test) – Comparison Group ...... 178 Table 43 Studywide Content Test (Pre-test to Post-test) – All Participants ...... 180 Table 44 Within Groups Studywide Content (Pre-test to Post-test) Score...... 181 Table 45 ATMI Post-treatment Scores by Scale and by Group...... 182 Table 46 Within Group Studywide ATMI Confidence Pre-treatment to Post-treatment Scores...... 183 Table 47 Within Group Studywide ATMI Value Pre-treatment to Post-treatment Scores...... 183 Table 48 Within Group Studywide ATMI Motivation Pre-treatment to Post-treatment Scores...... 184 Table 49 Within Group Studywide ATMI Enjoyment Pre-treatment to Post-treatment Scores...... 184

xv LIST OF FIGURES

Figure 1 Research Questions...... 61 Figure 2 Sequence of study activities...... 79 Figure 3 Studywide mathematics content test item addressing coordinate geometry ...... 87 Figure 4 Studywide mathematics content test item addressing inequality graphing ...... 88 Figure 5 Studywide mathematics content test item addressing variable manipulation ...... 88 Figure 6 Etch-a-Sketch digital toy model provided to students ...... 117 Figure 7 Student sketch showing aesthetic modifications for Etch-a-Sketch ...... 122 Figure 8 Student sketch showing mathematical modifications for Etch-a-Sketch ...... 123 Figure 9 Etch-a-Sketch featuring aesthetic and mathematical modifications ...... 127 Figure 10 Etch-a-Sketch, with modifications, as constructed by treatment student...... 128 Figure 11 Programming code for the penguin object ...... 129 Figure 12 Frogger video game model ...... 132 Figure 13 A Frogger sketch recorded in one student’s Initial Plans ...... 138 Figure 14 Frogger sketch showing planned design for a “Shark Drop” game...... 139 Figure 15 Completed “Lumberjack Attack” video game...... 144 Figure 16 Completed “Shark Drop” video game ...... 145 Figure 17 Tamagotchi Virtual Pet video game model ...... 147 Figure 18 Design plans for Tinki pet, including variables and aesthetic elements ...... 155 Figure 19 Planned Tamagotchi with multiple pets, scenes, interacting variables...... 156 Figure 20 Alien Tamagotchi Virtual Pet...... 162 Figure 21 World War II Battle (Tamagotchi Virtual Pet)...... 163 Figure 22 Bunny Tamagotchi...... 164 Figure 23 BanMan in the Jungle (Tagmagotchi Virtual Pet) ...... 165 Figure 24 Fluffo Sheep Tamagotchi Pet ...... 166 Figure 25 Completed Tinki Tamagotchi Pet...... 167

xvi CHAPTER 1

INTRODUCTION

This study addressed the mathematics that middle school students employed during their production of video games. It examined the mathematics content learned by

6th and 7th graders during the design and construction of three video game projects over the course of 7 months. The study also examined the ability of the student programmers to laterally transfer mathematics content learned during programming to the traditional context of a multiple-choice test. The performance of student programmers was compared with that of a control group of nonprogrammers on measures of mathematics content and attitude toward mathematics.

Background

Mathematics achievement has been identified as an area in which American precollege students are faltering (United States Department of Education, 2009).

Organizations including the National Council of Teachers of Mathematics (NCTM, 2000) and the White House (White House, 2009) are leading efforts at improvement. Emerging research has suggested that making mathematics instruction relevant and applicable in the lives of youth may impact math achievement, especially when it capitalizes on high- interest technologies such as video games (Kafai, Franke, Ching, & Shih, 1998; Ge,

Thomas, & Greene, 2006; Devlin, 2010).

Video Games

Video games are a regular part of daily life for young people. A 2010 report, "The

Kids and Games: What Boys and Girls are Playing Today," revealed that 91% of preteen

1 boys and 93% of preteen girls play games online (M2 Research, 2010). The widespread

interest of video gaming has resulted in $10.5 billion annual revenue from computer and

video games in the United States (Siwek, 2010), besting film and all other entertainment

venues. Preteens and teens spend more time gaming than on homework (Cummings &

Vanderwater, 2007), and parents and teachers debate whether schools could capitalize on

children’s enthusiasm for video games for an educational purpose (Rice, 2007).

When used to facilitate learning in instructional settings, video games can serve an

educational role. “Educational games and simulations are experiential exercises that

transport learners to another world. There they apply their knowledge, skills, and

strategies in the execution of their assigned roles” (Gredler, 1996, p. 571). Computer-

based educational games can address any academic subject from the liberal and

performing arts to science, technology, engineering, and mathematics (STEM). For

example, a game-based history software title is Muzzy Lane’s Making History, in which

students role play the leaders of nations embroiled in World War II, making political,

economic, and military decisions and interacting with other leaders to shape the direction

of the war. Similarly, an example from the STEM disciplines is Software Kids’ Time

Engineers, which invites students to travel in a time machine to three different eras and

encounter typical engineering problems to be solved, including building pyramids,

irrigating farm land, and operating medieval drawbridges. In general terms video games

have enjoyed little success in education because teachers typically do not see their

educational value (Korzeniowski, 2007). However, some schools are now beginning to

embrace the educational possibilities of video games, asking, “What if, instead of seeing

2 school the way we’ve known it, we saw it for what our children dreamed it might be: a

big, delicious video game?” (Corbett, 2010, p. 2).

Video Games and Mathematics

New attention is being focused on the use of technology-based solutions, including video games, for a particular instructional discipline deemed in need of substantial revision, mathematics education. This emphasis may be resulting from increased awareness of the need to address our nation’s comparatively poor student performance internationally in mathematics (United States Department of Education, 2009). Data collected during the most recent Trends in International Mathematics and Science Study

(TIMSS) (Gonzales et al., 2009) and the Program for International Student Assessment

(PISA) study (Fleischman, Hopstock, Pelczar, Shelley, & Xie, 2010), found that 4th and

8th grade students in the United States were being outperformed by peers in many advanced nations around the world, with that trend extending into high school.

Additionally, the national emphasis on improving mathematics instruction has been pushed by the rising expectations associated with the Elementary and Secondary

Education Act (US Dept. of Ed., 2010) and its accountability of schools, teachers and students through mathematics achievement testing in the new millennium.

An analysis by the National Mathematics Advisory Panel of major studies examining the use of instructional math software concluded that the use of such software in the classroom “has generally shown positive effects on students’ achievement in mathematics as compared with instruction that does not incorporate such technologies”

(US Dept. of Ed., 2008, p. 50). Much of the software on which this conclusion was based, however, focused on training knowledge-level understanding, not higher-order thinking

3 skills. But as new products are developed and research emerges regarding how these instructional software and video games can most effectively promote learning, the current emphasis of computer-assisted instruction (CAI) on tutorial and drill activities may diminish (Squire, 2005).

Looking beyond student engagement in commercially developed math software products may yield the most productive results. For example, the National Mathematics

Advisory Panel stated that, “studies show that teaching computer programming to students supports the development of particular mathematical concepts, applications, and problem solving” (US Dept. of Ed., 2008, p. 50). The Panel went on to note that, “research indicates that learning to write computer programs improves students’ performance compared to conventional instruction, with the greatest effects on understanding of concepts and applications, especially geometric concepts, and weaker effects on computation” (p. 51). This recommendation suggested that instead of students working as consumers of pre-made mathematics software, students should also be encouraged to use math in producing their own products, via computer software design and programming.

As a result of employing math in a programming capacity, lateral transfer (Wright, Rich,

& Leatham, 2010) of mathematics skills may occur from the computer lab to the mathematics classroom, reaching the overarching goal of boosting mathematics performance among students.

Youth-Oriented Programming Environments

Numerous programming environments exist for conducting computer- programming instruction in the precollege setting. For older teens, the selection of a programming language logically may be tied to the ability to extend experience in the

4 language to university-level and professional programming activities. Flash, C++, and

Java (i.e., the programming language required on the current advanced placement computer science exam) may be feasible choices at the high school level. But in K-8 schools, these languages are beyond the reach of child and preteen programmers. Instead, the ubiquitous Logo-based, “turtle” programming languages may provide more suitable, age-appropriate learning environments (Computer Science Teachers Association, 2006).

Selection of Programming Projects

Additionally, programming projects selected for implementation should be instructionally do-able and personally relevant to their intended audiences (Milbrandt,

1995; Computer Science Teachers Association, 2006). Do-able is defined as achievable and within reach of a student’s existing skill set in mathematics and logic, as well as ability to learn the selected programming language. Personally relevant is defined as engaging, interesting, and meaningful for the target age group, gender, and cultural identity. Projects in which students design their own video games and then write computer code to program the games for use by other students have been demonstrated as instructionally sound vehicles for teaching both mathematics and programming concepts

(Kafai, 1995). Even the President of the United States recognized the value of encouraging students to pursue the creation of their own games by announcing a competition that challenged, “middle schoolers to come up with a video game design that incorporates

STEM concepts and encourages learning in its areas of study” (Mitchell, 2010).

5 Attitudes Toward Mathematics

Finally, the role of attitudes toward mathematics cannot be divorced from investigations addressing achievement in mathematics. Studies have shown that attitude towards mathematics is clearly related to achievement in mathematics (Dwyer, 1993).

Dispositions must also be considered within the context of specific technologies used for instruction. As Squire (2005) pointed out, “while completion rates for online courses barely reach 50%, gamers spend hundreds of hours mastering games.” Determining mathematics dispositions of students designing video games may provide insight into possible motivational routes to mathematics instruction and elevated mathematics achievement.

Conceptual Underpinnings for the Study

In evaluating the teaching of mathematics through the vehicle of video game design and construction, four components of this problem must be addressed:

1. Recognizing and describing the mathematics inherent in the design of video games;

2. Learning complex tasks, specifically the task of computer programming a video game to create a functional product;

3. Lateral transfer of mathematics from a programming environment to traditional mathematical problem-solving tasks; and

4. Attitude of students towards mathematics and the role of attitude in achievement in mathematics.

6 Each component will be discussed briefly according to its own theoretical framework and in reference to its interconnection with the larger problem. Further, the problem will focus specifically on the middle school age group.

Recognizing Mathematics Inherent in the Design of Video Games

The ability to “recognize and apply mathematics in contexts outside of mathematics” defines the Connections principle of the NCTM Principles and Standards

(2000, p. 65). With regard to student design and construction of video games, students must be able to able to first recognize the mathematics inherent in a game prior to applying mathematics in the design and coding of their own original game.

Video games may be constructed through the application of multiple mathematical principles. Foundational knowledge in a breadth of topics – including counting, variable manipulation, coordinate geometry, conditional logic, and probability – is fundamental to and recognizing the mathematics of game play. Some examples of how these topics may be invoked include the following:

1. Counting may be applicable when determining how much time has elapsed or whether all the dots have been eaten in a Pac-Man game.

2. Variable manipulation may be involved in variety of game play activities, from decreasing a “lives” variable to adjusting a “health” variable of a game character.

3. Coordinate geometry may be needed to set the boundaries of the drawing pen in an Etch-a-Sketch digital toy, or to determine whether a collision has occurred in

Asteroids (i.e., when the spaceship and an asteroid share the same coordinates).

4. Conditional logic may be used in a game such as Frogger when determining if the frog is simultaneously in the stream and on a log – in which case he has not drowned.

7 5. Probability may be employed to produce a random number and provide a likelihood of obtaining an optimistic, pessimistic, or neutral fortune in a Magic Eight Ball video game.

In addition to proficiency in the above topics, ability in spatial reasoning is required in video games that entail movement in a virtual world onscreen. Spatial reasoning (i.e., mentally manipulating representations of objects) demonstrates the student’s ability to abstract game characters and interactions in place of their real-world analogues.

Students who demonstrate ability in spatial reasoning not only are more capable of recognizing spatial relationships in the video game context, but also have been shown to be more likely to major in STEM fields in university and ultimately pursue STEM careers

(Wai, Lubinski, & Benbow, 2009).

Recognizing and then representing the mathematics in an existing video game can be achieved only through the filter of mathematics already known to the student. For example, a student may recognize two balls in straight-line motion (e.g., one rolling horizontally with uniform motion and another falling vertically with accelerative motion) as having speed, but without knowledge of equations addressing gravitational acceleration, the student may not be able to represent mathematically how the two motions differ. The degree to which the student is capable of recognizing and representing the mathematics within a video game is limited by his knowledge of mathematics.

Even with an extensive mathematical skill set, a student still may not easily move from recognizing the math of a video game to representing it in the code of a programming language. Establishing a supportive instructional environment can foster the

8 making of such connections in the context of the video game. NCTM standards note that,

“challenging problems encourage students to think about how familiar concepts and procedures can be applied in new situations. New ideas surface quite naturally as extensions of previously learned mathematics. With prompting from their teacher, students routinely ask themselves, ‘How is this problem like what I have done before?

How is it different?’” (NCTM, 2000, p. 274)

Finally, the process of representing the mathematics of a video game may be achieved through multiple routes (NCTM, 2000). These include the use of (a) word representations, both oral and written, (b) mathematical symbols and formula representations, and (c) pseudocode or computer programming code representations.

Representations cannot be provided for a game in its entirety. Instead, representations define single states of objects or individual events (i.e., interactions between two characters or an interaction between a character and its environment) within a video game.

The combined effect of multiple object states and events constitutes a complete video game.

Learning Complex Tasks

From the design perspective, van Merriënboer (1990) viewed the video game as a complex entity comprised of smaller, component parts. This is because a video game results from the combined effect of hundreds (or more) objects interacting with each other and their environment. Playing a video game requires managing and mastering the component tasks of the game in the short run while simultaneously working purposefully towards the overarching, complex game goal in the long run. For example, short run tasks within the game Frogger include jumping through traffic and avoiding collision with

9 vehicles, while the long run task is to move five frogs safely to their lily pad homes.

Knowledge of the complex video game’s components defined by van Merriënboer (1990) is necessary for describing and ultimately constructing a video game.

From the literary perspective, Gee (2007) suggested that video games employ their own language and that characterizing a video game requires learning this unique language.

For example, “leveling up” carries both game play and mathematical meaning. From the game play perspective, the player has successfully lived to confront another round of obstacles in an environment of increased complexity; and from the mathematical perspective, the player’s “lives” count exceeds zero, his “health” is increased by an incremental value, and the “speed” at which he will be able to move is now doubled.

Knowledge of video game vocabulary defined by Gee (2007) is also necessary for describing and constructing a video game.

Moving students from recognizing to describing and ultimately constructing the mathematics of a video game must therefore be achieved with consideration both for the nature of complex tasks and for the language of the game.

Lateral Transfer of Mathematics

Papert (1998) was an early proponent of the idea that students may learn mathematics effectively not by playing video games, but instead by creating their own games. Kafai (1995) established the efficacy of this concept, showing that students building fraction games for peers increased math performance compared with non-game makers. However, it may be posited that the creation of any game – not just math games – inherently involves mathematics. Writing programming code to cause a virtual car to travel with constant motion, a ball to deflect from a paddle, or a score to rise

10 incrementally, requires mathematics. Students, operating as game makers, must invoke their knowledge of mathematics to create an onscreen world from abstractions born in their minds.

Unlike experimentation in the messy, real world, operations in a controlled computer microworld can be controlled and limited by the user. Fadjo, Hallman, Harris, and Black (2009) defined the experience of physically manipulating an agent designed to represent a particular object as surrogate embodiment, a mechanism that, “provides a unique pedagogical opportunity for the instruction of rudimentary arithmetic topics during video game design and development” (p. 2787). Thus, students building many different types of video games may ultimately increase their math performance by learning and applying mathematics in context.

Ferdig and Boyer (2007) also supported the benefits of game making over game playing, noting that, “one area within the video game arena that has received considerably less attention is the concept of student development of games. Educators [should] pay closer attention to student development of video games because it offers design experiences that can impact classroom learning.”

Because, “systematically pairing a core subject with another, complementary subject, may lead to greater overall learning in both subjects,” (Wright, Rich, & Leatham,

2010, p. 3529) students creating video games may engage in lateral transfer of math skills employed in game programming to other mathematical contexts. Thus, lateral transfer of mathematics learned in the context of video game programming may impact student mathematical achievement. This lent credence to the acceptability of offering students opportunities to learn mathematics in settings other than the traditional mathematics

11 classroom and in contexts beyond mathematics itself. Thus, it may be asked whether students learning mathematics in a video game programming environment are able to demonstrate similar performance levels on tests of mathematics achievement as students learning mathematics in a traditional math environment.

Attitudes Toward Math and the Role of Attitude in Achievement in Mathematics

Attitude is a general term expressing the affective dimensions of state of mind and the prevailing tendency of a person’s emotions, encompassing disposition, motivation, self-concept and other factors. Attitude is extremely important to academic success or failure in school, including achievement in mathematics. Tapia and Marsh (2004) summarized the work of Opachich and Kadijevich (2000) who investigated the role of attitude towards mathematics in math achievement as follows:

1. “Mathematics achievement is closely related to self-concepts and attitudes towards mathematics;

2. The effects of mathematics attitude on mathematics achievement is mediated by self-efficacy;

3. Confidence and self-esteem are linked at higher levels to success in problem- solving;

4. Confidence of success in a math-related course is a stronger predictor of choosing math majors than either confidence to solve mathematics problems or to perform math-related tasks” (p. 12).

Thus, understanding the role of student disposition towards mathematics may be equally as important as understanding true mathematical ability in evaluating academic success in math.

12 Because engagement in video games exerts a powerful affective pull on teenage audiences, exploring student attitude towards mathematics that is learned through video game creation may provide further insight to student achievement in mathematics.

Statement of the Problem

Mathematics competence in the United States is in crisis. American precollege students are underperforming peers of industrialized nations trailing 23 countries on standardized mathematics tests. Such results, “underscore concerns that too few U.S. students are prepared to become engineers, scientists and physicians, and that the country might lose ground to competitors” (Glod, 2007, p. A7).

Instructional technologies may prove helpful in mitigating deficiencies in mathematics among students (US Dept. of Ed., 2008). The National Mathematics

Advisory Panel of the United States Department of Education, “recommends that high- quality computer assisted instruction (CAI) drill and practice, implemented with fidelity, be considered as a useful tool in developing students’ automaticity (i.e., fast, accurate, and effortless performance on computation), freeing working memory so that attention can be directed to the more complicated aspects of complex tasks” (US Dept. of Ed., 2008, p. 51).

The Panel also recommended that computer programming be considered as an effective tool (US Dept. of Ed., 2008) in teaching mathematics, especially when programming languages such as Logo are employed.

While computer programming is not typically included in K-8 curricula, many high schools do offer advanced placement courses in the subject (Computer Science

Teachers Association, 2006). However, postponing programming instruction past middle

13 school may impact not only technology fluency, but also mathematical literacy. The

National Research Council (1999) stated that a basic understanding of computer science, including programming, is now an essential component for preparing high school graduates for life in the 21st Century.

Providing opportunities for students to engage in programming opportunities prior to high school may, “be best accomplished by adding short modules to existing science, mathematics, and social studies units” (Computer Science Teachers Association, 2006, pp.

10-11). Because mathematics requires a grasp of computation, algebraic manipulation, algorithmic thinking, problem solving, and troubleshooting – the same tasks required for computer programming – teachers may be able to capitalize on the natural connection between the two disciplines.

Teachers of elementary and middle grades may also find that motivating students to work on mathematics and programming problems is more effective when students engage in tasks they find interesting and relevant to their world. An analysis of the TIMSS assessment found that students who demonstrated positive attitudes towards mathematics were more likely to perform well in it (Mullis, Martin, Gonzalez, & Chrostowski, 2004;

Mullis, Martin, & Foy, 2008). The popularity of video games, especially among preteens and young teens may provide fruitful content for engaging this audience – not for game playing, but for game making. Unlike mundane real-world problems about which students may care little (e.g., computing sales tax), video games may also appeal to student interests and provide a motivational impetus for achievement in mathematics.

Thus, as summarized by Squire (2003), “Given the pervasive influence of video games on American culture, many educators have taken an interest in what the effects

14 these games have on players, and how some of the motivating aspects of video games might be harnessed” (p. 2). To obtain a measure of these achievement effects and motivational aspects, a study is proposed in which middle-grade students employ mathematics and Logo-based programming software to design functional video games.

Evaluating achievement in mathematics and disposition towards mathematics in the course of the study may provide evidence of whether or not classroom instruction in video game design correlates with cognitive and affective gains in these domains.

Purpose of the Study

This research studied the efficacy of employing a technology-based, alternative curriculum to convey middle school mathematics content and foster affective satisfaction among students. The target audience consisted of preteen and teen youth in middle school.

Research Questions

Three research questions guided this study, addressing the mathematics of video game design and construction, the transfer of math ability, and disposition toward mathematics.

The mathematics of video game design and construction. What mathematics content do middle school students invoke as they design and construct video games? This question entailed three parts: (a) analysis – What mathematics content do middle school students invoke as they analyze games? (b) synthesis – What mathematics do middle school students invoke as they synthesize games? (c) programming – What mathematics do middle school students invoke as they program games?

15 The transfer of mathematics ability. Using a standards-based multiple-choice mathematics content test, two questions were posed: (a) Does the performance of middle school students who engage in video game design and construction (analysis and synthesis) improve after creating video games? And (b) Does the performance of middle school students who engage in video game design and construction (analysis and synthesis) exceed the performance of students of similar math abilities who are not engaged in creating video games?

Attitudes toward mathematics. Does the attitude of middle school students towards mathematics improve after designing video games, and how do these dispositions compare with students who are not engaged in video game design?

By examining the three research questions longitudinally with an audience of middle school students, insight into the success of employing alternate methods in teaching and learning mathematics was sought.

Hypotheses

Three hypotheses were posited to address the research questions under investigation in this study. Each of these question follows.

Hypothesis regarding the mathematics of video game design and construction.

With regard to the first research question, it was hypothesized that middle school students who engaged in video game design and construction would learn age-appropriate mathematics concepts (e.g., measurement, geometry and algebra) as prescribed by standards outlined by the National Council of Teachers of Mathematics. Learning was measured via (a) evaluations of mathematical events written as students analyzed video

16 games, (b) evaluations of representations written as students synthesized video games, and

Hypothesis regarding the transfer of mathematics ability. With regard to the second research question, it was hypothesized that (a) that students engaging in video game design and construction would improve performance, pre-project to post-project, on a test of standards-based mathematics content ; and (b) that students engaging in video game design and construction would achieve higher score gains on a studywide test of standards-based mathematics content than peers of similar mathematics abilities who did not engage in video game design an construction as compared on pre- and post-treatment administrations of the test.

Hypothesis regarding attitudes toward mathematics. With regard to the third research question, it was hypothesized that (a) within the treatment group, attitudes towards mathematics would increase from pre- to post-treatment measurements; and (b) that post-treatment attitude scores towards mathematics would be higher for treatment group students than for comparison group students.

Definitions

Several key terms are used throughout the study. Their definitions follow.

Analysis. “Analysis examines a problem and splits it up into its components”

(Kafai, 1995, p. 7).

Design. “A process of problem-solving. Finding a solution that satisfies the given conditions,.., involving processes such as planning, search, decision making, reasoning, and management of mental resources. In the design process, two phases are distinguished:

17 analysis and synthesis. In both analysis and synthesis, designers develop strategies, such as modularization or generating alternative solutions, when dealing with complex design tasks” (Kafai, 1995, p. 7).

Digital Toy. Digital toys are similar to video games in terms of appearance and interaction. However, digital toys differ from video games in that they possess no overarching goal (Wolf, 2000). Designing and constructing digital toys offers a simplified entry point for the design and construction of video games. Digital toys, including digital

Etch-a-Sketches, digital Magic 8-Balls, and digital paper dolls, can feature a limited number of mathematical and programming concepts for their function when compared with video games. This is due, in part, to the fact that scoring, collision detection, or boundary recognition are usually not required for digital toys – a digital toy is essentially a subset of a video game (Gee, 2007). In this study, the Etch-a-Sketch project is representative of a digital toy, while Frogger and Tamagotchi Virtual Pet are video games.

However, for the purposes of the current study, “video game” typically will be used to encompass both “video games” and “digital toys” except where otherwise noted.

Elements. Video game elements are visual components of the game play field.

Elements include characters, obstacles, score tallies, and background scenery. Elements may move and interact (e.g., characters) or remain static (e.g., scenery).

Events. Events are interactions that define game play and cumulatively determine success or failure of the goals. Events may include moving a character, the collision of two objects, the collection of an object, and the scoring of a point. Table 1 shows example game events specified for three video games, along with their associated content standards defined by the National Council of Teachers of Mathematics (NCTM, 2000).

18 Table 1

Example Video Game Events and Associated NCTM Content Standards

Video game project Video game event Mathematics content standard

Constrain tool tip from Write inequalities to describe Etch-a-Sketch drawing in the red frame boundaries (algebra)

Reset frog’s starting Use coordinate geometry Frogger coordinates following a (geometry) collision with an obstacle

Scale dimensions of the Write algebraic equalities to Tamagotchi Virtual Tamagotchi to reflect values manipulate variables (algebra Pet of its health/hunger variables and measurement)

Event representations. Event representations in this research consist of verbal descriptions (words), mathematical formulations (symbols, logical statements), and computer programming code or pseudocode. Table 2 shows representations for an example event pertaining to the Tamagotchi Virtual Pet video game.

Table 2

Example Event Representations in Tamagotchi Virtual Pet

Representations Event Mathematical Programming

Put Feed procedure in Feed button Feed the If the hunger variable is Tamagotchi greater than 1, then to feed pet to decrease subtract 1 from the current if hunger > 0 his hunger value of the hunger. [sethunger hunger – 1] end

19 Goals. Video game goals are the target benchmarks that the player seeks to accomplish during game play. A long-run goal, such as hopping a frog through a busy intersection without incident in Frogger, consists of several short-run goals, such as avoiding traffic and completing the crossing before time expires.

Synthesis. “Synthesis focuses on bringing all the different parts together to a solution” (Kafai, 1995, p. 7).

Video game. “An audiovisual entertainment whose content is largely representational” (Wolf, 2000, p. 2). “Games played on game platforms (such as the Sony

Playstation 2 or 3, the Nintendo Game Cube or Wii, and Microsoft’s Xbox or Xbox 360, or various handheld devices) and games played on computers” (Gee, 2007, p. 1). In the current research, Frogger and Tamagotchi Virtual Pet are video games that students design and construct. Frogger is representative of the dodging genre of video games, while

Tamagotchi Virtual Pet is representative of the artificial life genre (Wolf, 2000).

Video game analysis. Examining an existing video game; analyzing game components to recognize elements, goals, and events.

Video game construction. Translating representations for game elements and events into a functional video game. This entailed writing computer programming code.

Video game synthesis. Representing game events in multiple formats, including both mathematical and programming.

20 CHAPTER 2

REVIEW OF RELATED LITERATURE

There is something odd about the way we teach mathematics in our schools. We

teach it as if we expect that our students will never have occasion to make new

mathematics. We do not teach language that way. If we did, students would never

be required to write an original piece of prose or poetry. We would simply require

them to recognize and appreciate the great pieces of language of the past, the

literacy equivalents of the Pythagorean theorem and the law of Cosines. (Schwartz

& Yerushalmy, 1987, p. 293)

Introduction

Learning mathematics through a nontraditional route such as video game construction is representative of the type of “new mathematics” characterized by Schwartz and Yerushalmy (1987). In light of the novelty of this nontraditional approach, it is unsurprising to find only a limited volume of research that directly informs new investigations addressing video game construction and its correlation with mathematical achievement and affect. Nonetheless, each research question in the current study can be examined with regard to theory and previously conducted studies regarding component facets of each question including the language of video game design, precollege student programming, the significance of mathematics achievement, mathematics instruction in nontraditional contexts, transfer of mathematical content and processes, and affect and achievement in mathematics.

21 Literature Addressing the Mathematics of Video Games

Examining literature relevant to the mathematics of video construction entailed four components: (a) providing a rationale for the video game context, situating it from both social and educational perspectives; (b) the theory of video game design and construction; (c) programming environment options appropriate to the ability of preteenagers; and (d) teaching mathematics via video game construction.

Rationale for the Video Game Context

Video games have emerged as a popular and ubiquitous form of entertainment, especially among pre-teenagers and young teens. Reception of the video gaming phenomenon by educators and parents has ranged from embracing this form of tele- mediated engagement as the next generation of learning to rejecting it as a mindless waste of time (Johnson, 2005). However, recent research and the success of schools such as the progressive Quest to Learn (Q2L) secondary institution in New York – with its entire curriculum built on a serious games and video game development platform – have strengthened the perspective that gaming holds real educational value (Hsu, 2010).

Learning theorists and educational researchers have posited that participation in video gaming results in more than simple “increased eye-hand” coordination, contending that global thinking and complex reasoning skills are acquired by the player-learner (Gee,

2007; McGonigal, 2011). These complex skills emerge not only from video game playing, but also video game construction, thereby offering a rationale for students to migrate from game consumers to game producers (Papert, 1998). As one team of game researchers noted, “with so many children and adolescents playing video games, one might surmise that the task of creating a game has a sort of ‘universal’ appeal that could be utilized to

22 explore various academic topics through video game design and development” (Fadjo,

Chang, Hong, & Black, 2010, p. 2674).

The announcement by President Obama of a nationwide, video game design challenge, inviting middle school youth to build and compete their own original games, further strengthens the emerging perspective that game-building promotes learning in a way that traditional methods no longer achieve (Mitchell, 2010). New learning environments are being explored for ways in which video game construction serve instructional purposes that depart profoundly – and effectively – from textbook and pencil realms (Kearney, 2004; Kelleher & Pausch, 2005; Maloney, Peppler, Kafai, Resnick, &

Rusk, 2008; Klopfer, Jenkins, & Perry, 2010). The “open-ended learning environment

(OLE)” that fosters student game development is contrasted against the “decontexualized” environment of the traditional classroom (Ge, Thomas, and Green, 2006), differentiating the problem-based setting where students learn content to solve the larger problem (i.e., create a video game) from the classical setting where mastering content is an end in itself

(Jonassen, 1999).

Video games can address any subject matter, academic and otherwise, and students constructing their own games can showcase everything from social studies to sports in their game themes. However, the processes of designing a game concept and ultimately bringing that concept to fruition requires a very specific skill set predicated on the underlying theory of video game design and construction.

Theories of Video Game Design and Construction

Constructing a video game requires the complex composition of a virtual world that presents challenges for players to engage in while there. It is simultaneously a

23 creative, artistic, mathematical, logical, and technological process. With genres ranging from arcade to adventure (Wolf, 2000), the resulting product can be compellingly fun and enticing because playing it engages us in ways that go beyond reading a book or even playing a board game.

Two key researchers, linguist James Paul Gee and experimental psychologist and instructional technologist Jeroen van Merriënboer have delineated the symbiotic components associated with video game creation: first, the “design grammar” defining the microworld aesthetics and game play of a video game; and second, the “complex tasks” associated with constructing the game in a computer language.

Video game design: contributions of James Gee. Defining video gaming as its own “semiotic domain,” Gee has examined how, similar to other distinct disciplines including genetics or opera, gaming possesses its own design grammar, namely its own content (e.g., facts, theories, and principles) and social practices (Gee, 2007). Mastering a video game is an exercise in obtaining fluency in gaming design grammar, but acquiring video game literacy is not an isolated endeavor with relevance only to the confines of the computer or video game console. In fact, “the principles on which video game design is based are foundational to the kind of learning that enables children to become innovators and lifelong learners” (Gee, 2005b, p. 3).

To this end, Gee has enumerated thirty-six such learning principles in What Video games Have to Teach Us About Learning and Literacy (2007), examining their roles both with regard to general learning and, more specifically, to gaming. While Gee’s main interests relate to game playing as opposed to game making, he does argue strongly in favor of the latter’s constructionist aspect of video gaming: “What is most powerful about

24 video games… is that the consumer (player, learner) is also a producer. They can fairly easily build extensions and modifications to many games” (Gee, 2003, p. B17). This production concept can be explored not only in the context of video game modding (i.e., creating new or altered content to be shared via the Web) but also in the more extended context of complete game design by the learner.

Several of Gee’s 36 video game learning principles are especially germane to mathematical problem solving by students engaged in video game design. For example, the design principle encompasses the overarching philosophy of why students should be encouraged to engage in video game design. This principle states that, “learning about and coming to appreciate design and design principles is core to the learning experience” (Gee,

2007, p. 41). When designing their own video games, learners must analyze existing games first in order to know what defines the game – what constitutes its elements, goals and events – before they can advance to game synthesis. Gee’s insider principle encompasses the constructionist perspective of video game design, stating that “good learning requires that learners feel like active agents (producers) not just passive recipients

(consumers)” (Gee, 2005a, p. 6). By engaging in active investigation to produce understanding, students learn the design grammar, including the mathematics of video game design, and can apply that understanding in the production of something new.

With regard to the process of using a programming language to synthesize working games on the computer, Gee’s amplification of input principle holds special relevance:

“For a little input, learners get a lot of output” (Gee, 2007, p. 64). Unlike using a protractor, straightedge and calculator to compute and draw a motion path, learners constructing a video game with a mindtool such as Logo can quickly test math and physics

25 concepts without getting bogged down in the mechanics of computations and graphical plots.

Logo is also critical to facilitating learning with regard to Gee’s subset principle, which states that “Learning even at its start takes place in a (simplified) subset of the real domain” (Gee, 2007, p. 141). Creating a microworld in Logo affords learners the opportunity to model precisely such a subset. “The real world is a complex place. Real scientists do not go out unaided to study it. Models are all simplifications of reality and initial models are usually ‘fish tanks,’ [that is] simple systems that display the workings of some major variables” (Gee, 2005a, p. 12). When designing new video games, learners start small and increase in complexity as they gain competence in the process. However, starting small does not mean that basic skills are learned in isolation or out of context.

Gee’s bottom-up basic skills principle requires that learners understand how individual tasks are situated in context of the entire game. “ When learners fail to have a feeling for the whole system which they are studying, when they fail to see it as a set of complex interactions and relationships, each fact and isolated element they memorize for their tests is meaningless” (Gee, 2005a, p. 14).

Gee suggested that these contextualized basic skills must be practiced repeatedly for the learner to gain fluency and confidence in their use. The concentrated sample principle advises that, “the learner sees, especially early on, more instances of fundamental signs and actions than would be the case in a less controlled sample. Learners get to practice them often and learn them well” (Gee, 2007, p. 142). For example, in the early stages of video game construction, learners analyze and program repeated instances of basic motion and heading in simple games devoid of collisions, scoring and other

26 complex constructs. Practicing smaller tasks builds to task fluency that is ultimately applied to production of the entire game.

To assist learners in building firm foundations in video game design, teachers must not only provide opportunities for repeated practice of concepts and skills (concentrated sample principle), but also carefully sequence the acquisition of new concepts and skills.

“The order in which learners confront problems in a problem space is important” (Gee,

2005a, p. 9). This idea embodies the problem-solving principle which states that, “learning situations are ordered in the early stages so that earlier cases lead to generalizations that are fruitful for later cases” (Gee, 2007, p. 142). A teacher experienced in video game design process can provide key guidance in moving the learner along from simple to related but more complex concepts and skills. “Generalizations are best recognized by those who already know how to look at the domain, know how the complex variables at play in the domain relate and interrelate to each other” (Gee, 2005a, p. 10).

Once new concepts and skills are acquired during video game design, learners can solidify their understanding through practice and then use this new learning in novel situations, including new games. This embodies Gee’s transfer principle, namely that

“learners are given ample opportunity to practice, and support for, transferring what they have learned earlier to later problems” (Gee, 2007, p. 142). For example learning to make a car drive down a road requires the design and coding of straight-line, horizontal motion.

Once secure, the learner can transfer this concept to creating straight-line motion for any action such a ball rolling or a dog walking.

Bringing a complete video game to fruition, students follow similar paths as one another in terms of analyzing the game environment and then synthesizing a working

27 program. But these paths need not be identical, bringing Gee’s multiple routes principle to bear: “There are multiple ways to make progress or move ahead. This allows learners to make choices, rely on their own strengths and styles of learning and problem-solving”

(Gee, 2007, p. 105). Learners designing games do not have to follow a single path towards a finished product. Unlike solving a series of mathematics problems on a worksheet, students designing video games do not always end up with the same answers. For their game to work, the answers must be correct, just not necessarily identical.

Gee suggested that video gaming and video game construction, “if implemented in schools, would necessitate significant changes in the structure and nature of formal schooling as we have long known it, changes that may eventually be inevitable anyway given modern technologies” (Gee, 2005a). Thus, researching how video gaming and principles of game making impact student learning – specifically preteen and teen acquisition of mathematical problem-solving skills – may result in new ways of thinking about mathematics curriculum and instruction in the middle school.

Video game construction: the contributions of Jeroen van Merriënboer. The mechanics of video game construction may be modeled on instruction of analogous complex tasks as described by van Merriënboer. Such instruction, states van Merriënboer must, “take both human cognitive architecture and multimedia principles into account to ensure that learners will work in an environment that is goal-effective, efficient, and appealing” (van Merriënboer & Kester, 2005, p. 72).

Van Merriënboer characterized the learning of complex tasks including mathematical problem solving and computer programming as hierarchical. To describe this process, van Merriënboer established the 4C/ID (four component instructional design)

28 model of complex learning. This model, “explicitly aims at the integration of knowledge, skills and attitudes; the ability to coordinate qualitatively different constituent skills; and the transfer of what is leaned to daily life or work settings” (van Merriënboer & Kester,

2005, p. 72). The model is designed to provide instruction that avoids bombarding the learner with excessive information (Sweller, van Merriënboer, & Paas, 1998). Reducing the cognitive load of learners by focusing their efforts in a scaled-down and well- controlled model correlates with Gee’s subset principle (2007), the idea that learning should transpire in a simplified subset of the real domain.

The 4C/ID model describes the composition and sequence of an entire learning task. Applying the 4C/ID model to video game design and construction, 4C/ID defines the tasks and sequence via which the teacher guides learners in analyzing, synthesizing, and programming an entire game. In terms of Gee’s learning principles (2007), 4C/ID invokes the problem-solving principle by delineating an ordered sequence of individual tasks to be practiced, mastered and combined to create larger project task.

The 4C/ID is predicated on four components that function as a fluid, yet cohesive whole during the instructional process. These components follow.

1. Learning tasks component. Each individual learning task represents the lowest level of decomposition of the entire complex task. Individual learning tasks are grouped into larger tasks, which are then sequenced to comprise an overarching complex task, such as constructing a complete a video game. Building fluency in a task progresses from extensive support from the teacher when learners first encounter a new task to no support from the teacher once learners are proficient at executing the task independently (van

Merriënboer & Kester, 2005).

29 Complete, worked examples may serve as helpful models for students first learning a task (Paas & van Merriënboer 1994). For example, a student first attempting to write programming code to move an object called “racecar” forward by 10 pixels in a game must be provided a complete programming command (e.g., RACECAR, FD 10) in order to learn the proper syntax of the language (van Merriënboer & Krammer, 1987).

Completion tasks (which present a problem and a partial solution with gaps which the learner must complete) are appropriate support tools for students who have achieved a moderate level of task proficiency following exploration of worked examples. For example, a student attempting to write programming code to move an object called “frog” forward by 10 pixels in game could be provided a partial line of programming code (e.g.,

FROG, ? ) with the expectation of completing the missing syntactical information (van

Merriënboer & Krammer, 1987).

2. Supportive information component. The learner is provided context for the overall task via case studies, systematic approaches to problem solving and cognitive feedback. This information can be viewed as the theoretical backbone of the overall task.

In the context of video game construction, students would learn tasks (Component

1) in the context of case studies addressing the specific games in which those tasks would be implemented. For example, students planning to build a Frogger video game could examine a commercial version of Frogger to evaluate the game play (i.e., analysis).

Discourse between teachers and students regarding the implementation of each task in the game (problem-solving) would constitute cognitive feedback.

3. Procedural information component. Procedural information is essentially

“how-to” instructions or demonstrations, provided as-needed, and paired with corrective

30 feedback. It provides “algorithmic specification of how to perform those routine aspects”

(van Merriënboer & Kester, 2005, p. 72) of individual learning tasks.

Video game construction requires the development of procedures, groups of programming commands executed together. Writing such procedures usually requires how-to guidance in thinking algorithmically about the selection and sequence of this information.

4. Part-task practice component. Practice takes the form of additional exercises that help the learner ramp up to automaticity in executing the individual learning tasks.

Learners’ repetitive practice of similar tasks embodies Gee’s concentrated sample principle of learning (Gee, 2007).

For example, students designing a video game featuring several objects moving in the playfield will work to set the directional heading and motion for each object. This repeated process will build automaticity with regard to the specific actions these commands perform.

Research has shown the 4C/ID model to be effective in training learners to perform complex tasks ranging from air-traffic control to writing computer programs (van

Merriënboer & Kester, 2005) including the programming of geometrical concepts

(Sweller, van Merriënboer & Pass, 1996). This same model of instructional design may also prove successful in teaching video game construction to middle schoolers.

Programming Environment Options for Preteenagers

Programming environments are fundamental to the creation of video games. While graphic design and animation software is vital to creating the imagery and feel of the game

“world,” game action cannot come to life without computer programming code and the

31 underlying mathematics, physics, and logic required to write the code.

Although video gaming, multimedia, and technology as a whole are exploding in society and the economy, preparation of students in the computing disciplines, including programming, is actually receding in the K-12 arena. “The point of engaging youth in computer programming is not to turn them all into hackers or programmers, but because being engaged in the full range of technology fluencies–including programming–is an educational right of the 21st Century” (Maloney, Peppler, Kafai, Resnick, & Rusk, 2008, p. 370 ). Nonetheless, computer science instruction, which includes algorithmic and computational thinking as well as programming, is not required in K-12 schools. And schools that do offer computer science courses count them as elective credits, with fewer than 10 states granting math or science credit for their completion (Wilson, Sudol,

Stephenson, & Stehlik, 2010).

A wide variety of programming languages and environments are available at the precollege level with little standardization regarding which programming choice is most appropriate within each age band. To organize the many programming options, Kelleher and Pausch (2005) created a taxonomy that profiles this diversity of offerings, organizing languages by their goals. They noted that, in all languages, “to successfully write a program, users must understand several topics: how to express instructions to the computer (e.g., syntax), how to organize these instructions [structure] and how the computer executes these statements” (Kelleher & Pausch, 2005 p. 86). The taxonomy ranges from environments that simply empower the user (e.g., The Incredible Machine), to teaching languages that emphasize structure over syntax (e.g., Scratch), and others that require users to learn both syntax and structure (e.g., MicroWorlds EX).

32 At the high school level, the Java programming language is used on the Advanced

Placement exam (College Board, 2010). At the K-8 level, options including Alice, Scratch,

Terrapin Logo, and Stagecast have emerged as successful tools for teaching computer programming (Redden, 2007). Newly released environments including GameStar

Mechanic and Scratch are encouraging the next generation to create their own games

(Salen, 2007; Gee, Hayes, Torres, Games, & Squire, 2008; Peppler & Kafai, 2007). Unlike professional programming languages, most youth-oriented programming environments offer drag-and-drop interfaces, drop-down menus, interlocking “puzzle piece” commands, and other friendly adaptations to ease the challenging work of the novice programmer. As

Kelleher and Paucsh (2005) noted, these inviting interfaces create a more accessible starting point “by simplifying the mechanics of programming, by providing support for learners, and by providing students with motivation to learn to program” (p. 131).

However, research needs to be conducted regarding the success students experience in transitioning from programming in these “teaching environments” to programming in

“professional environments” in high school and as post-secondary, adult learners.

One class of programming environments, Logo and its derivatives, offers greater transparency to the underlying mathematics of programming because it retains more of the traditional coding interface than other teaching environments. Specifically, MicroWorlds

EX, a modern version of the Logo language originally developed more than 30 years ago by Seymour Papert at MIT, retains the fidelity of not only teaching programming but teaching children mathematics – Papert’s fundamental goal. Logo is representative of the constructionist philosophy of learning in which students construct not only their own

33 ideas, but also physical objects, including computer programs, demonstrative of those ideas (Papert, 1999).

The Logo environment appears to the learner as a two-dimensional world in which objects called “turtles” can be placed in the world and manipulated through simple commands. But what Logo offers the learner is far more powerful. “Children can identify with the Turtle and are thus able to bring their knowledge about their bodies and how they move into the work of learning formal [mathematics]” (Papert, 1993, p. 56).

Unlike traditional means of parsing knowledge in a fact-based framework, Logo- produced microworlds offer learners, “a context for the construction of ‘wrong’ (or, rather,

‘transitional’) theories” (Papert, 1993, p. 56). Learners can create and test models to see how they perform, then easily adjust model parameters and add new commands to enhance and extend their creations. As an example, a simple, introductory activity in Logo involves the construction of familiar geometric figures (e.g., triangles, squares, pentagons.). The learner gives the turtle iterative commands to generate each regular polygon, employing trial-and-error to compute the appropriate turning angles – and eventually deducing the relationship between the number of sides and the turning angle for each polygon. Although the same activity can be conducted via pencil and paper drawings, the use of Logo as a mindtool shifts the focus from the mechanics of construction to the concepts of construction by reducing cognitive load in the learner (Sweller, van

Merriënboer, & Paas, 1998).

In the MicroWorlds EX environment, learners move from writing computer programs to constructing figures to working in a virtual environment that they can control, experiment in, and craft models which can be shared with others. While Logo, and more

34 specifically MicroWorlds EX, are not intended to be catch-all solutions for teaching mathematics and its applications, they do serve as examples of a successful adaptation of the immersive, learn-by-doing environment espoused by constructionist theorists – one particularly well-suited to the study of applied mathematics.

In Papert’s view, school mathematics, “often means a lonely, impersonal experience of manipulating symbols in accordance with rules learned by rote” (Papert,

1975, paragraph 10). Logo was Papert’s attempt to mitigate negative perceptions of math by providing learners a computer-based microworld that they can control, experiment in, and share models with others. “Logo programming… is clearly an effective medium for providing mathematics experiences… when students are able to experiment with mathematics in varied representations, active involvement becomes the basis for their understanding. This is particularly true in geometry – and the concept of variable”

(McCoy, 1996, p. 443).

Of the many ways Logo can be explored, Papert said, “my favorite example is having kids learn enough programming [to make] their own video games. Almost all kids find this exciting… because video games are important in their world” (Papert, 1996, paragraph 19). Papert noted that, while game-playing and game-describing are starting points in fostering mathematical thinking, mindtools like Logo can facilitate learners’ metamorphoses from game consumers to game producers. “When they get the support and have access to suitable software systems, children’s enthusiasm for playing games easily gives rise to an enthusiasm for making them, and this in turn leads to more sophisticated thinking about all aspects of games” (Papert, 1998, paragraph 13).

35 Since the authentic process of making a game requires the learner, working as game designer and programmer, to set characters in motion (represent velocity), collide with or capture objects (register coordinate convergence), and record scores (change variable values), the learner must inevitably employ mathematics in the production of even the most basic games. “But if you are going to make a serious video game, you are very likely to run into mathematical problems. For example, if you take a jump, how do you describe the trajectory? You need a mathematical concept for describing it, a parabola” (Papert, 1996, paragraph 19). Thus, learners will likely find a natural and real progression from the application of simple to complex mathematics as they build skills and extend their abilities in creating ever more robust video games. As a programming language that may be used to craft simple video games and is easily learned by middle school students, MicroWorlds EX is a logical choice for the environment to study learner acquisition of mathematics skills through game building.

Focus on Math Instruction in the Video Game Construction Context

“Most schools use technology to teach content and only few offer opportunities to learn programming” (Maloney et al., 2008, p. 367). Thus, a considerable body of research has examined student learning in various school subjects through multimedia technology, including some addressing serious video games, but only a few studies have looked at programming per se and even fewer have looked at mathematics instruction as relates to programming. Now, with national attention focused on the failings of STEM education, especially mathematics education, (US Dept. of Ed., 2009), further research examining instruction of mathematics through the closely related programming context is merited.

And of the many projects programming instruction can be employed to create, video

36 games are some of the more personally relevant to students in the preteenage and teenage age group (Teske & Fristoe, 2010).

Some research conducted on the mathematics students that learn as a result of programming video games has focused on student development of games that explain or teach mathematics content. For example, Kafai’s extensive work detailed in Minds In Play

(1995) profiles the construction that preteenage students performed in their creation of

“Fraction-World” Logo games intended to serve as instructional software teaching fraction concepts to younger peers. Kafai found that students who migrate from playing games to constructing games, performing both initial design and subsequent software programming tasks, invoke deep thinking and generate rich, high-level schema in their production activities.

Ultimately, though, the more relevant mathematics students employ in the construction of video games may instead be inherent in their design and programming of any game – arcade games, adventure games and sports games – not just games that specifically teach mathematics. For example, in a longitudinal study of preteenagers and teenagers programming animated scenes and games addressing a wide variety of themes, not specifically “math,” Maloney et al. (2008) found that students demonstrated significant year-to-year gains in employing the programming concepts of loops, Boolean logic, variables, random numbers, and conditional statements – all of which have direct mathematical corollaries. And Fadjo, Chang, Hong & Black (2010) are currently investigating preteen acquisition of expertise in manipulating conditional statements – concepts fundamental to mathematical proof – through the “embodied” (i.e., concrete) approach of Scratch’s puzzle programming tiles.

37 As students design and construct their own games, they access and apply their mathematical understanding during two distinct phases: analysis and synthesis (Kafai,

1995). By Kafai’s definition, “analysis examines a problem and splits it up into its components, whereas synthesis focuses on bringing all the different parts together to a solution” (1995, p. 7). For students to successfully analyze an existing video game and synthesize their own version of the game in a computer-programming environment, they must employ both a deep understanding of mathematical problem solving and the ability to apply their understanding in the game context.

Literature Relevant to the Transfer of Mathematics Ability

Beyond examining whether learners can recognize (analyze), represent

(synthesize), and computer program (construct) mathematics concepts in the production of a video game, consideration must be given as to whether learners can transfer successful employment of mathematics in video game construction to other mathematics contexts.

While many other mathematical contexts exist, one such context that has received significant attention is traditional paper-and-pencil problem solving as measured by achievement tests (Mullis, Martin, & Foy, 2008).

In considering the process of mathematical transfer, we must first situate (a) the overall significance of student mathematical achievement; (b) the mathematics standards of middle school math courses; (c) instructional methods in mathematics; and finally, (d) the nature of lateral transfer from a related subject area to mathematics.

38 The Significance of Measuring Mathematical Achievement

Precollege student achievement in mathematics is a national concern, impacting national prosperity and economic health (Rouse, 2010). Government organizations and educational researchers routinely measure mathematics achievement among precollege students to gauge the readiness of emerging generations to innovate in science, technology, engineering, and mathematics (STEM) disciplines.

Results from the 2009 Nation’s Report Card in Mathematics (US Dept. of Ed) showed that only 39% of American fourth graders and 34% of eighth graders perform at or above the proficient level (US Dept. of Ed., 2009). Reports including the Third

International Mathematics and Science Study (TIMSS, by Mullis, Martin, & Foy, 2007) indicated that not only are American children not meeting internal benchmarks, but the

United States is being outperformed by many other industrialized nations as measured on mathematics achievement tests administered in the middle grades, and further, that our nation is not improving its performance (Gonzales et al., 2009). “The United States ranks much worse than most of our economic competitors in the mathematics performance of high school students. That national slide begins in middle school” (Devlin, 2010).

Because of the red flags signaled by the national downslide in mathematical performance, and because middle school has been identified as a critical time period when student learning begins to derail (Gonzales et al., 2009), efforts are underway to improve mathematics instruction in order to close the achievement gap in the preteen and early teen years. Such efforts address one or more factors of mathematics instruction believed to be associated with student achievement, including teacher preparation, instructional methodology, technology integration, assessment and student disposition.

39 Mathematics Standards in the Middle Grades

Over the past decade, national mathematics education organizations have compiled keystone documents including Principles and Standards for School Mathematics by the

National Council of Teachers of Mathematics (NCTM, 2000); Adding It Up by the

National Academies (Kilpatrick, Swafford, & Findell, 2001), Foundations for Success

(US Dept. of Ed., 2008), and the Common Core Standards (Common Core State

Standards Initiative, 2010) that outline the body of knowledge and skills that students who are successful in mathematics know. The documents “reflect research that has identified the importance of learning with understanding, as opposed to memorizing isolated facts, and the importance of opportunities to engage in mathematical reasoning and problem solving” (National Research Council, 2010, p.108).

School curricula and textbooks vying for lucrative, state adoption are typically written around NCTM’s Principles and Standards (2000), a document that details the content and processes that students should learn from grades K through 12. These standards define the mathematics middle schoolers should learn to be prepared for further mathematical study in high school and beyond, with “ambitious expectations are identified in algebra and geometry that would stretch the middle-grades program beyond a preoccupation with number” (NCTM, p. 211).

Mathematics content standards. NCTM content standards describe the mathematics knowledge students in the middle grades should posses and consist of the standards that follow.

40 Numbers and operations standard. This standard includes understanding numbers

(their representations and their relative relationships); understanding operations; and computing and estimating fluently.

Algebra standard. This standard includes understanding patterns, relations and functions; using algebraic symbols (i.e., variables) to represent and analyze mathematical situations and structures; using mathematical models to describe quantitative relationships; and analyzing change.

Geometry standard. This standard includes analyzing properties of two- dimensional and three-dimensional geometric figures and describing geometrical relationships in those figures (i.e., the Pythagorean Theorem); specifying spatial locations and spatial relationships (i.e., parallel and perpendicular lines) via Cartesian coordinates and other representations; applying transformations and recognizing symmetry; and using

“visualization, spatial reasoning, and geometric modeling to solve problems” (NCTM,

2000, p. 233) including applying geometric ideas in contexts beyond the mathematics classroom.

Measurement standard. This standard includes understanding measurable attributes (i.e., angles) as well as measurement units and systems; and selecting and applying formulas to compute geometric attributes (i.e., using the area formula in a circle and solving rate of motion questions).

Data analysis and probability standard. This standard includes asking questions to which data collection and statistical analysis can answer, then inferring and predicting based on the data; and understanding and applying probability concepts.

41 Role of the content standards in video game design and construction. Each of the previous content standards plays a role in the context of video game design and construction. Students require numbers and operations to describe quantities such as distances in the game environment; computation such as addition and subtraction is required to increase or decrease quantities in response to game interactions. Algebra understanding is required to set and manipulate variables. Geometry is necessary to set the starting position of objects in the play field as well as to create parallel motion of multiple objects. A grasp of measurement is fundamental to setting the heading of game objects and determining the area of zones of interaction in which collisions between objects may occur. Finally, data analysis and probability is needed to understand the relative frequency with which certain game events may occur as well as introduce a degree of randomness into object interactions.

Mathematics process standards. NCTM process standards for the middle grades explain how students should acquire and use mathematical content outlined in the content standards. The five process standards follow.

Problem-solving standard. Problem solving entails the integration of multiple math topics while “engaging in a task for which the solution method is not known in advance,” in contexts that include, “familiar experiences involving students' lives”

(NCTM, 2000, p. 52). As students problem solve, they “engage in larger problems, perhaps occasionally working for several days on a single problem and its extensions”

(NCTM, 2000, p. 256).

Reasoning and proof. “Students at all grade levels should learn to investigate their conjectures using concrete materials, calculators and other tools” (NCTM, 2000, p. 57).

42 “Young children's explanations will be in their own language and often will be represented verbally or with objects. Students can learn to articulate their reasoning by presenting their thinking to their groups, their classmates, and to others outside the classroom” (NCTM,

2000, p. 57).

Communication. Students should employ oral and written communication to convey their mathematical thinking and practice using the language of mathematics.

Connections. All students should “recognize and apply mathematics in contexts outside of mathematics” (NCTM, 2000, p. 274).

Representations standard. All students should exhibit “use of representations to model physical, social, and mathematical phenomena” (NCTM, 2000, p. 70).

Role of the process standards in video game design and construction. Within the context of video game design and construction, extensive problem solving is required as multiple mathematics content topics must be incorporated in the creation of each game, and each game is a “large problem” that necessitates several days (or more) to solve

(NCTM, 2000). Students constructing video games employ reasoning and proof (Van

Lehn, 1983), using the computer microworld as a visual tool to demonstrate their conjectures and manifest their ideas for internal and external (peer-to-peer) discussion.

Those discussions exemplify the communications standard (NCTM, 2000). The representations process standard is employed when students (a) write verbal, mathematical, and programming code representations during video game synthesis, and

(b) produce their functional game artifact that features a virtual world comprised of characters and events representing people, objects, behaviors, and events observed in real- world contexts (NCTM, 2000). Students working to apply mathematics in a programming

43 environment to create the game world are inherently employing the connections standard by using their math skills in a format that moves beyond the paper-and-pencil problem set format (Kafai, 1995).

Curricula and Instructional Methods in Mathematics

Conducting studies that control for confounding variables to provide direct comparisons between diverse curriculum and varied instructional methods have proven difficult. Thus, consensus does not exist regarding which curriculum and instruction are most successful in increasing student mathematical achievement (Confrey, 2006). There is some agreement that part of the problem stems from teaching conducted in the middle grades, in part due to differences in how middle school math is leveled; in some states grades five through eight are grouped with elementary levels and in others these grades are grouped with secondary levels. The National Academies (National Research Council,

2010) have addressed the particular challenge associated with training teachers to instruct mathematics in the middle grades, noting that, “preparation for middle grades mathematics teachers varies from place to place, and certification requirements reflect the ambiguous status of middle school” (p. 103).

While standards (e.g., NCTM’s Principles and Standards, 2000, and the Common

Core State Standards, 2010) guide teachers in knowing what students mathematics content should learn and how students should proceed when problem-solving in mathematics, specific methods of mathematics instruction vary widely. The diversity of these methods, from algorithmic to discovery-based, and from pencil-and-paper to technology-infused, complicate the instructional landscape. In the end, outcomes of student mathematical

44 achievement may provide some indication of the degree to which various instructional methods succeed.

For example, the Increasing Course Rigor project, a multistate, collaborative initiative by the National Governors Association Center for Best Practices and ACT, Inc. focused on training teachers to use state-of-the-art curriculum units and new instructional methods that were integrated with a system of assessments to, “prepare more high school graduates for the demands of higher education and the workplace” (Wakelyn, 2008, p. 1).

In one strand of this initiative, success was measured according to how well geometry teachers tightened their alignment with mathematics content standards as determined by the resulting performance of students on the ACT, a standardized, college entrance exam.

Some studies have been tied to specific curriculum offerings from textbook publishers, but offer insight to empirical practice. For example, Saxon Math mathematics demonstrated the success of its “incremental, distributed” instructional approach characterized by a cycle of instruction, practice, and assessment of small chunks of information (Williams, 1986). However, other researchers examining similar, traditional approaches found that while children can learn and successfully employ computational algorithms, they do so without depth of understanding (Brown & Burton, 1978; Van Lehn

1983) and such approaches negatively affect future learning (Hiebert, 1984; Baroody &

Ginsburg, 1986).

Conversely, researchers examining the efficacy of the Everyday Mathematics program of instruction (the instruction received in daily math courses by participants in the current study) found that students benefited from learning through this nontraditional curriculum and instruction. Everyday Mathematics focuses on a cyclical approach,

45 covering a diverse range of topics in each lesson, and cycling back from month-to-month and year-to-year with increased depth in each topic. Students engaged in Everyday

Mathematics were found to be successful at inventing their own methods for computation

(Kamii & Joseph, 1988; Resnick, Lesgold, & Bill, 1990; Carpenter, Fennema, & Franke,

1992) and these methods bolstered their skills in estimation and mental calculations. Other field-based findings, though, indicated that the lack of rote drill-and-practice in Everyday

Mathematics reduced student acquisition of automaticity required for executing multi-digit multiplication and division, and negatively impacted student ability to excel in future mathematics (Clavel, 2003).

More beneficial to an examination of mathematics instructional methods may be findings discussed in the research-based, How Students Learn, History, Mathematics, and

Science in the Classroom (Donovan & Bransford, Eds., 2005). How Students Learn outlines three fundamental principles of learning theory that teachers must invoke in the instruction of mathematics, regardless of the specific curriculum in place in the classroom.

Principle 1 requires connecting with and building on the prior mathematical understandings that students bring to the classroom. Principle 2 involves relating the procedural skills and competences required to do mathematics to an overarching conceptual framework. Principle 3 aids students in invoking their own metacognitive strategies when solving mathematics problems.

These principles resonate with Gee’s problem-solving learning principle and van

Merriënboer’s 4C/ID approach to teaching complex skills in that all three describe a cycle of connecting with prior knowledge, relating within the overarching framework, and reflecting metacognitively about solution strategies. Thus, the parallel processes

46 demonstrate a natural relationship among game playing, game making and mathematical problem solving.

Lateral Transfer from a Related Subject Area to Mathematics

A fundamental question to be investigated is whether video game construction projects in the middle school classroom provide a meaningful context for teachers to enable learners “to recognize and apply mathematics in contexts outside of mathematics”

(connections standard, NCTM, 2000). Research and theory can inform such engagement; however, few studies examining the efficacy of the reverse, learning mathematics through a context (i.e., video game design and construction), have been conducted.

Perkins and Salomon (1987) addressed this concept of learning in a primary field having spillover into learning in a secondary field. They noted that, “transfer of learning is a side effect, a fringe benefit: You learn A and find that performance B improves as well… You learn one video game and note that the reflexes developed there help you with another similar game” (p. 287). Transfer of learning may be positive or negative depending upon the degree of “interference effects between the vocabulary and syntax of the original and new language” (Perkins and Salomon, 1987, p. 287), for example, between the language of computer programming and the language of mathematics.

Further, transfer may be horizontal (lateral), as in the case of learning computer programming contributing to learning mathematics; or vertical, as in the case of learning a task that is a component of a larger task (i.e., learning to subtract contributes to learning long division). Another aspect of transfer is distance. Near transfer occurs between two tasks that are similar in character and context (i.e., driving a car and driving a truck), while far transfer occurs between two tasks that share few common aspects in character and

47 context (i.e., computing probability and playing craps). A final component of transfer is type, low road and high road. “Low road transfer occurs as the automatic consequence of varied practice.” (Perkins and Salomon, 1987, p. 288). In contrast, high road transfer takes place, “through the mindful abstraction and application of principles” (Perkins &

Salomon, 1987, p. 290).

With regard to the current research, learning tasks consist of representing and constructing video games in three forms: verbal, mathematical, and programming. As students analyze the structure of provided game models and then construct their own versions of each game, they move among these three forms, employing a series of near transfer and far transfer tasks that includes both load road and high road aspects.

Analyzing the mathematics inherent in a video game to enumerate events as verbal descriptions is a far transfer task as it shares only some common aspects of character and context with the visual nature of the game. Synthesizing representations in mathematical and programming forms is also far transfer in terms as it entails translation from verbal to computational, logic, and code forms. Analysis and synthesis can also be considered low road because students mostly write events and representations that relate closely to the examples provided given by the instructor (Perkins & Salomon, 1987). Programming the mathematics and programming representations into a new video game may include both near transfer tasks (i.e., that share many common aspects with the representations elicited in the model game) and far transfer tasks (i.e., that show significant variation or combine multiple examples ideas from the model) (Perkins & Salomon, 1987). For example, some programming tasks require repetition or slight variation of a simple piece of code (e.g., using the SETH command to set the headings of objects to point different directions in the

48 play field); these tasks are essentially near transfer in nature. Other tasks require an understanding of multiple tasks variations that result in related but dissimilar code (e.g., setting the heading of an object to a random direction following a collision, using a combination of SETH, RANDOM, and OnTouching code), requiring far transfer learning. Game programming may also be low road or high road depending on the degree of deviation students exhibit in creating their games. Students who program a game nearly identical to the game model, but varying only in aesthetic or slight substantive ways engage in low-road transfer. Students who choose to deviate from the game model and create new game features that are not simply aesthetic in nature engage in high-road transfer. This is because they must apply previously learned principles in a novel fashion, or possibly learn new principles in order to implement their ideas (Perkins & Salomon,

1987).

Beyond understanding the mathematics that students invoke in analyzing and synthesizing video games, one goal of the current research is to explore the lateral transfer of mathematics expertise in the programming context to mathematics expertise in the traditional context. Singley and Anderson (1989) have shown that through specifically designed instructional environments and related hands-on projects, learners not only can acquire new knowledge but they can transfer that knowledge for use in other applications.

Creating a game exemplifies Wilensky’s (1991) notion of concretizing a concept, specifically transforming mathematical concepts into a physical product. Additionally, creating a video game requires the additional step of programming code to execute game play in a computer environment. Fadjo et al., (2010) have termed this idea of concretizing

49 abstract instructions in a physical game program that is manipulable and executable as embodiment.

With regard to the programming context, van Merriënboer (1990) found that highly structured programming instruction including worked examples and varied practice tasks builds not only the successful programming skills in the learner, but also the ability to transfer these skills to novel situations (van Merriënboer & Kester, 2005). Because

"people construct new knowledge and understandings based on what they already know and believe" (Bransford, Brown, & Cocking, 1999, p. 10), it is plausible to expect that some degree of transfer will occur from students exhibiting mathematical understanding in the design and programming of games to mathematical problem-solving in other familiar contexts – potentially answering problems on achievement-style math tests.

Klahr and Carver (1988) explored whether children can learn high-level thinking skills from computer programming and whether these skills transfer to other, similar environments. They found that learners with extensive experience in Logo effectively learned to troubleshoot errors in nonprogramming, multistep written solutions. Detecting and correcting errors, whether “plugging in” a value to check the solution for a variable or tracing the logic of a geometry proof, are skills necessary in all forms of mathematical problem solving. Other research showed similar findings, including those by De Corte

(1992) who examined learners’ debugging strategies, problem decomposition, and construction of external representations, finding that, “the Logo treatment was successful in achieving transfer, especially to situations that are not too different from the original learning context” (p. 326).

Fadjo, Chang, Hong, and Black (2010) promoted an approach known as embodied

50 cognition to investigate the processes by which preteen programmers learn one specific aspect of writing code that possesses strong ties to mathematics, that of conditionals.

Common programming conditional statements, such as “if-then” and “if-then-else” are often paired with logical connectors (e.g., “and,” “but,” and “or”) to facilitate decision- making in the program. Similar constructs are also utilized in writing proofs in first-year geometry courses. Because of the abstract nature of writing conditionals, it is posited that embodying these programming statements with a physical form (e.g., a puzzle tile in

Scratch) in a video game may more effectively allow students to recognize, “when a condition is met and when the consequence sequence should be executed” (Fadjo, Chang,

Hong, & Black, p. 2672).

While problem-solving skills will likely grow organically as learners’ experiences in a programming environment such as Scratch or MicroWorlds EX develop, Littlefield et al., (1988) emphasized maintaining an instructivist (i.e., teacher-guided) approach during some phases of Logo instruction to establish the basis for effective skills transfer. He noted specifically that the instructor must provide, “explicit emphasis on problem-solving skills, as well as explicit bridging to decontextualize the skills developed and encourage transfer to other contexts” (p. 133).

Most encouraging that students enhance mathematical understanding through programming may be Kafai’s findings (1995) that preteens and young teens taught Logo in a game design context scored higher on rational number concept post-tests than control groups. The concept tests assessed learners’ understanding of fractions translating among pictorial, written, and symbolic representations. Additionally, “the Game Design Class showed a significant increase in their ability to deal with fractions” (p. 266) on a

51 standardized test subsection focused on algorithmic operations, rising from 55% (pre-test) to 70% (post-test) following Logo intervention.

One limitation associated with studies of Logo’s efficacy in promoting the acquisition of mathematics concepts and skills is that many studies were conducted in the late 1970’s, coinciding with the original release of Logo and the initial widespread adoption of Logo for use on early school computer labs. Little new research has been conducted on Logo-derivatives during the past decade, although a few studies (Fadjo,

Chang, Hong, & Black, 2010; Maloney, Peppler, Kafai, Resnick, & Rusk, 2008) have recently revisited this realm through the vehicle of Scratch. Only the work of Kafai and her colleagues (Kafai, Franke, Ching, & Shih, 1998) looked specifically at learning mathematics through the MicroWorlds EX interface.

Kelleher and Pausch (2005) emphasized that programming is not an isolated learning experience but offers students the opportunity “to explore ideas in other subject areas” (p. 84). Resnick and Silverman (2005) extended this belief, noting that, “When kids learn to program, it extends the range of what they can design, create, and invent with the computer. Moreover, it provides them with experience in using and manipulating formal systems – experience that is important not only in computer science but also in many other domains from mathematics to grammar to law” (p. 199). But as Teske and Fristoe (2010) observed, “The shape of this transfer between subject area domains, however, has remained unexplored and unformulated” (p. 166) including the transfer from doing mathematics in the context of constructing a video game to doing mathematics in the traditional paper-and-pencil context. Thus, in summarizing the state of research in the field of student-programmed video games, it has been observed that, “Taking a design

52 approach to researching games might provide a useful framework for studying games, which thus far, have lacked a coherent research paradigm” (Squire, 2003, p. 11).

Literature Relevant to Student Affect Associated with Mathematics

In addition to instructional methods, mathematical performance exhibits a relationship with student attitudes towards mathematics. Understanding how affect, including attitudes towards participating in instructional activities, math self-efficacy, enjoyment of mathematics, and other measures correlate with student achievement in mathematics may better inform educators, curriculum developers, and other decision- makers regarding instructional alternatives that lead to student success in mathematics.

Although more attention than ever is being directed toward mathematics and mathematics achievement, little of that attention includes an affective component. As Chamberlin noted, “unfortunately, dispositions and motivation are the components of education that are potentially the items most frequently neglected as a result of increased attention to standardized assessments” (2010, p. 167).

Examining mathematics and affective relationships may be especially important when considering a diversion from more traditional, instructional environments in favor of more constructivist, project-based activities such as video game construction, in which students must invest significant effort to produce a finished product (Blumenfeld et al.,

1991). These types of projects, “can increase student interest because they involve students in solving authentic problems, in working with others, and in building real solutions (artifacts). Projects have the potential to enhance deep understanding because students need to acquire and apply information, concepts and principles, and they have the potential to improve competence in thinking because students need to formulate plans,

53 track progress and evaluate solutions" (Blumenfeld et al., 1991, pp. 372-373). If students are more likely to succeed in learning mathematics when engaged in learning activities they enjoy, then such relationships must be studied and demonstrated to provide justification for deviation from traditional instructional materials and methods in the school setting.

Definition of Affect

Affect is a term that encompasses a wide range of aspects characterizing an individual’s emotional response. Affect includes attitudes, disposition, motivation, feelings, beliefs, and emotions (Chamberlin, 2010).

The lack of a standard definition for affect has not impacted research efforts in attempting to measure it within student populations, especially with regard to mathematics teaching and learning. However, the development of instruments to quantify affect is predicated on identifying its measurable components. One definition developed by

Anderson and Bourke (2000) incorporated most key components of affect by stating that it is, “comprised of the sub-components: anxiety, aspiration(s), attitude, interest, locus of control, self-efficacy, self-esteem, and value” (Chamberlin, 2010, p. 168). External factors including peer and parental influences, however, are not explicitly included in this definition.

Each affective component also possesses three characteristics that must be considered in fully defining it, specifically target, intensity and direction. “The target refers to the object, activity, or idea towards which the feeling is directed. The intensity refers to the degree or strength of the feelings. The direction refers to the positive or negative orientation of feelings” (Chamberlin, 2010, p. 170).

54 Instruments for Measuring Affect

Measuring affect in mathematics requires an instrument that may be administered or protocol that may be implemented in order to quantify affective components. McLeod

(1994) observed that such instruments were employed as early as the 1960s for the purpose of determining student attitudes towards a new curriculum that taught mathematics through science. However, “one major concern of research on attitude was the quality of the instruments that were being used” (McLeod, 1994, p. 638).

Historical affective instruments. Many of the earliest instruments focused on single dimensions of affect and provided very narrow windows on the attribute they intended to study. Representative unidimensional instruments included the Dutton Scale, which measured students’ feelings toward arithmetic (Dutton, 1954); and the Attitudes

Toward Mathematics survey (Gladstone, Deal, & Drevdahl, 1960). Aikin (1974), was one of the first researchers to develop a multidimensional attitude survey; his instrument included both Enjoyment of Mathematics and Value of Mathematics scales, while

Sandman created a six-attribute scale, the Mathematics Attitude Inventory (1980), that added (a) perception of the mathematics teacher, (b) anxiety toward mathematics, (c) self- concept in mathematics, and (d) motivation in mathematics to Aiken's two scales. Other multidimensional scales were developed by Michaels and Forsyth (1977) as well as

Haladyna, Shaughnessy, and Shaughnessy (1983) who created tools for measuring teacher quality and learning environment variables, proposing that these components were actually more likely to impact class mentality towards mathematics than individual attributes. The Trends in International Mathematics and Science Studies (TIMSS) reports

(Mullis, Martin, & Foy, 2007) featured three affective scales consisting of the Index of

55 students’ positive affect toward mathematics (PATM), the index of students valuing mathematics (SVM), and the index of students’ self-confidence in learning mathematics

(SCM). The Fennema-Sherman Mathematics Attitude Scales (Fennema & Sherman, 1976) consists of nine instruments: attitude toward success in mathematics scale, confidence in learning mathematics scale, mathematics anxiety scale, effectance motivation scale in mathematics, mathematics as a male domain scale, mother scale, father scale, teacher scale, and mathematics usefulness scale. Although the Fennema-Sherman is one of the most popular and frequently used sets of measurement scales, Mulhern and Rae (1998) have argued that the instrument actually addresses fewer factors than it purports to.

Ultimately, Fennema-Sherman may best be used for its measure of four scales, namely attitude, self-efficacy, motivation, and anxiety (Chamberlin, 2010).

Criteria for distinguishing quality affective instruments. To distinguish quality affective measurement instruments, Chamberlin (2010) suggested three criteria.

1. Statistical data. Instruments with high validity and reliability coefficients exceeding 0.80 are considered sound (Nunnaly, 1978).

2. Innovation. Instruments that measure some new aspect of affect are rated as innovative;

3. Level of usage in the field of mathematics education. Instruments used in multiple studies and cited in literature reviews possess a high level of use.

One additional practical consideration for an affective instrument intended for students is that administration time is minimized (Tapia & Marsh, 2004).

56 Attitudes Toward Mathematics Inventory (ATMI). One recently developed instrument for the purpose of measuring affective factors associated with learning mathematics that meets the aforementioned criteria is the Attitudes Toward Mathematics

Inventory (ATMI) by Tapia and Marsh (2004). The ATMI is a 40 item-instrument with demonstrated content validity, reliability and test-retest stability. The ATMI possesses content validity as it relates each of the 40 question items to the four affective variables of self-confidence, value, enjoyment, and motivation; it features a strong internal consistency and reliability of scores (item-to-total correlation) with a Cronbach alpha of 0.97 for the entire group of 40 items; and its Pearson coefficient for test-retest reliability for the total scale is 0.89, indicating that scores are stable over time. (Tapia & Marsh, 2004). The

ATMI has been used successfully with elementary, secondary and college audiences

(Fengfeng, 2006, 2007; Marsh, 2005; Moldavan, 2007) and has been employed in recent years in studies examining mathematics affect and achievement.

Affect and Achievement

Affect and achievement in mathematics are intertwined such that examining either component in isolation risks failure in obtaining a complete picture of student learning in mathematics. “Affect is a large piece of why students perform as they do … [it] is arguably the single greatest factor that impacts the learning process” (Chamberlin, 2010, p. 169).

Correlations between affect and achievement have been explored in numerous studies (Burstein, 1992; Reynolds & Walberg, 1992; Randel, Stevenson, & Witruk, 2000;

Sanchez & Sanchez-Rhode, 2003); by the Canadian government (Bussière, Cartwright, &

Knighton, 2004); and by developers of affective measurement instruments such as the

57 ATMI (Tapia and Marsh, 2004). Correlations have also been examined longitudinally via the regularly administered international TIMMS studies (Mullis, Martin, & Foy, 2008).

Featuring three independent, Likert-response scales, TIMMS has gathered affective data on precollege students worldwide on indices of students’ positive affect toward mathematics (PATM); students’ valuing mathematics (SVM); and students’ self- confidence in learning mathematics (SCM). At both the fourth and eighth grade testing stages, high achievement in tested mathematics content was associated with more positive affects, while low achievement scores were associated with more negative affects.

The positive correlation between mathematics achievement and affect also appears to display some causal directedness as well. A long held perspective that, “student beliefs and attitudes have the potential to either facilitate or inhibit learning” (Yara, 2009, p. 336) implies that attitude affects achievement. However, research by Ma & Xu (2004) indicated that the causal relationship may be the converse – that achievement in mathematics is a predictor of affect towards mathematics. Their examination of data from the Longitudinal

Study of American Youth (LSAY, 2007) showed that among students in grades 7-12, prior achievement played a significant role in predicting later attitude, but prior attitude did not predict later mathematics achievement (Ma & Xu, 2004).

While the results obtained by Ma and Xu (2004) implied that performing poorly in mathematics leads to negative attitudes towards mathematics, efforts are still being directed at improving math attitudes as a component of overall mathematical success among students. In fact, “developing positive attitudes toward mathematics is an important goal of the mathematics curriculum in many countries” (Mullis, Martin, & Foy, 2008, p.

173). An additional possibility regarding causality between affect and achievement is that

58 the relational is actually cyclical as suggested by Gibbons, Kimmel and O’Shea (1997). In this case, students’ attitudes may both lead to and stem from achievement in both positive and negative reinforcing loops. Thus, “those students who do well in a subject generally have more positive attitudes towards that subject and those who have more positive attitudes towards a subject tend to perform better in that subject” (Yara, 2009, p. 338).

With regard to examining students’ mathematics affect in conjunction with mathematics achievement measured during their construction of video games, findings may provide insight regarding mathematics interest and acquisition both in the short term and the long term. Not only can such data inform practice regarding alternative routes to teaching mathematics, but it may also offer insight to the larger challenge of engaging more youth in STEM-related careers. Hembree (1990) noted that when otherwise capable students possess negative affect toward mathematics, their resulting avoidance of studying mathematics reduces career options and deteriorates our nation’s human resource base in

STEM fields. Thus, if learning experiences involving content perceived as high-interest among preteenagers, specifically video game construction, generate positive affective and achievement gains in mathematics, the impact may be realized well beyond any one student or individual classroom.

59 CHAPTER 3

METHODOLOGY

The overarching question for the current study examined whether video game design and construction by middle school students is an instructionally effective and affectively satisfying method of teaching age-appropriate, standards-based mathematics content and processes. In exploring this larger proposition, this study addressed three different, yet related questions.

The first question involved the mathematics content and processes of video game construction by students. This question included three parts, namely video game analysis, video game synthesis and video game construction. Video game analysis focused on the mathematics that students utilize in recognizing events that occur during game play. Video game synthesis addressed the mathematics that students utilize in representing events that occur during game play. Video game construction examined the mathematics that students employ as they computer program actual games.

The second question sought to determine if the level of mathematical content knowledge students demonstrated prior to and following video game design and construction was improved. It also sought to determine whether students engaged in video game design and construction would achieve higher levels of mathematical content knowledge than students of similar math abilities who did not engage in video game design and construction.

The third question sought to go beyond the cognitive aspects of mathematics and video games by quantifying the affective dimension of middle school students towards mathematics. Attitudes of students who designed and constructed video games were

60 compared prior to and following project engagement. Additionally, it was queried how attitudes compared between students who engaged in video game design and construction and those who did not.

Research Questions

The specific research questions that guided this study follow (see Figure 1).

Figure 1. Research Questions.

61 1. The mathematics of video game design and construction. What mathematics content do middle school students invoke as they design and construct video games? This question entailed three parts: (a) analysis – What mathematics content do middle school students invoke as they analyze games? (b) synthesis – What mathematics do middle school students invoke as they synthesize games? (c) programming – What mathematics do middle school students invoke as they program games?

2. Lateral transfer of mathematics ability. Using a standards-based multiple-choice mathematics content test, how does the performance of middle school students who engage in video game design and construction (a) change, pre-design and post- construction of video games? And (b) how does it compare with the performance of students of similar math abilities who are not engaged in video game design and construction?

3. Attitudes toward mathematics. How can the attitudes of middle school students towards mathematics be characterized prior to constructing video games and after constructing video games, and how do these attitudes compare with students not engaged in video game design and construction?

Research questions were examined via multiple research methods, including both quantitative and qualitative analyses.

Research Design

The research study was designed based on a scientific design philosophy. Design methods included both quantitative and qualitative measurements.

Design Philosophy

The National Research Council (National Research Council, 2002), a science and technology advisory council of the federal government, set forth three guiding questions with regard to designing and conducting scientific research in education, asking specifically (a) What is Happening? (b) Is There a Systematic Effect? and (c) Why or How is it Happening?

Posing research questions, designing research methodologies, and interpreting findings in accordance with these guiding questions helps ensure relevance of the study to broader educational issues, replicability of the findings, and implementation of strategies and interventions judged successful in the educational setting.

With regard to specific mathematics content to be addressed in the middle grades, the National Council of Teachers of Mathematics (NCTM) provides content standards in its Principles and Standards for School Mathematics (NCTM, 2000). “Standards are descriptions of what mathematics instruction should enable students to know and do. They specify the understanding, knowledge, and skills that students should acquire” (p. 30).

NCTM content standards addressing (a) numbers and operations, (b) algebra, (c) geometry, (d) measurement, and (e) data analysis and probability prescribe specific content that students should learn in their mathematics studies in school.

Research questions and the video game design and construction projects developed to explore those questions were aligned with NCTM (2000) content standards. Alignment was accomplished by examining grade-appropriate standards, then creating game projects and mathematics content test questions based only on those standards. This ensured relevance to examination of student achievement of mathematics in the middle school.

63 This research project followed the NRC research principles and the NCTM national standards for mathematics content by examining instruction of mathematics in a context meaningful to students (Midbrandt, 1995). Specifically, the study described the acquisition and application of mathematics content knowledge through the processes of video game design and construction (Kafai, 1995). The research also compared the attitudes between students who constructed video games and students who did not. The research was designed and executed as a hybrid descriptive and quasi-experimental study.

The quasi-experimental design of the study prevented drawing conclusions regarding causal effects between video game treatment and student performance in mathematics ability and disposition. However, the potential for demonstrating correlational relationships existed, insofar as there was accounting, “for influential contextual factors within the process of inquiry and in understanding the extent to which findings can be generalized” (NRC, 2002, p. 80). A review of various paradigms and achievement measures led to the decision to use both quantitative and qualitative methodologies in conducting the study. In similar research involving construction of video games by middle school students, Kafai (1995) suggested an integrated approach featuring a diversity of research methods, “to compensate for the shortcomings of one methodology with the strengths of the other” (p. 38). This suggestion was followed in selecting the research methodologies for the current study.

Overview of Quantitative Methods

Quantitative research methods employ systematic scientific investigation of properties and phenomena and are useful when numerical descriptions are sought in explaining revealed relationships. Quantitative methods used in this study included

64 methods used for all study participants and methods used for treatment group students only. These methods consisted of the following.

Standardized tests in mathematics. The collection and assessment of standardized test scores. Annual achievement tests that existed for all study participants were use as the measure of comparison. These tests were used to determine pre-study, baseline mathematics abilities of the entire study population.

Studywide mathematics content tests (pre/post study). A multiple-choice mathematics content test designed specifically for this research project was administered to all study participants pre-treatment and post-treatment. The studywide mathematics content tests determined pre-treatment and post-treatment mathematics content knowledge on a specific set of mathematics content standards relevant to the construction of video games addressed in the study.

Checkpoint mathematics content tests (pre/post video game project). A multiple-choice mathematics content test designed specifically for each video game project was administered to treatment group participants pre-analysis and post- construction of each video game project. Scores on checkpoint mathematics content were used to determine incremental changes in mathematics content knowledge relevant to the construction of each specific video game.

Event-recognition tallies. During the analysis phase of each of the three video games, students were asked to write an initial list of the events they recognized while examining the game models. A simple count of the number of events recorded by each treatment group student was tallied for each video game.

65 Representations scores. During the analysis phase of each of the three video games, a rating of representations written by treatment group students for each event was established. A rating of -1, 0, or +1 was used to evaluate the depth and accuracy of mathematical and programming code representations recorded.

Game model events included. During the construction phase of each of the three video games, students were ask to construct games that included, at a minimum, all events featured in the game model. Game model event inclusion tallies recorded the number of events featured in the game model that students incorporated in their own games. While most events were both mathematical and programming in nature, some events featured only one type of representation.

Modifications included. During the construction phase of each of the three video games, students had the option of modifying, enhancing, and adding new events to their games. Modification inclusion tallies recorded the number of alterations and amplifications students included in their games as they brought their unique ideas to fruition.

Attitude inventory scores (pre/post treatment). Attitudes toward mathematics scores were collected from responses by all study participants on the ATMI attitude survey. The Likert-scale attitude survey was conducted pre-treatment and post-treatment.

Overview of Qualitative Methods

Interpretation of the collected qualitative data was accomplished within the context of the specific setting, participants, activities and instruments that comprised the study.

Qualitative data gathered throughout the study elucidated relationships that appeared in the quantitative data but required further corroboration to acknowledge or eliminate

66 explanations for the results. Such contextual considerations were, “especially critical for understanding the extent to which theories and finding may be generalizable to other times, places, and populations” (NRC, 2002, p. 5). Qualitative methods used in this study were employed only with treatment group students and consisted of the following.

Self-reflections (pre/post video game project). Documented by treatment group students, self-reflections were generated in the form of written notes and drawings on pre- formatted reflections and plans templates. Students recorded their self-reflections twice for each game: first, between the synthesis and construction phase of each game project

(i.e., as Initial Reflections and Design Plans); and second, following game construction

(i.e., as Final Reflections). Self-reflections included students’ perceptions of their challenges and successes, design graphics of their video games, and comments regarding modifications or deviations from the game models that students planned and implemented.

Observations. Data was also collected from observations of treatment students and their work activities that transpired during each class meeting. The researcher recorded field notes, in the form of observational notes and direct transcriptions of student statements, to the extent that such recordings did not infringe upon classroom instructional time. Researcher head notes were recorded in written form immediately following each class to capture additional nuances of classroom activities. Observation notes were used to corroborate student-written reflections and video game files.

Talk-alouds. Talk-alouds were conducted at least once per game project, following final construction of the video game and more frequently as time permitted.

Conducted by the researcher, digital audio recordings (and subsequent written transcriptions) were made in which individual students talked about their video game

67 construction activities. The intent of the talk-alouds was to obtain additional insight into students’ mathematical concept understanding during video game design and construction

(Ericsson & Simon, 1993; Patton, 2002). Occasionally, whole room talk-alouds were recorded to capture the dynamics of social interaction among students as they constructed their games. Additionally, to situate, “individual students’ progress in a larger context… case studies [were used] to illuminate the general trends” (Kafai, 1995, p. 38) if the stories of specific students in the treatment group proved compelling.

Evaluation of completed video games. Completed video games were evaluated for mathematical and aesthetic elements. Video game elements included those defined by the basic video game models as well as those added as modifications or deviations from the game models.

Quantitative Methods Regarding Research Question 1a: Video Game Analysis

A quantitative methodology was appropriate for examining Research Question 1a which focused on the mathematics of video game analysis. Data from assessment of student game design journals employed during the analysis phase of each game project were scored quantitatively to obtain measures of student content knowledge in mathematics during the analysis phase of each video game. The specific quantitative measure was event-recognition tallies.

Event-recognition tallies. Following presentation of a game model, treatment students were asked to make a simple count of identifiable events in the game. For example, in an Etch-a-Sketch toy, a student might have identified and named an event,

“move the pen tip.” In terms of the four-component/instructional design theory (van

Merriënboer & Kester, 2005), which states that complex tasks can best be taught via

68 carefully sequenced and practiced sub-tasks, this event would be considered a task which could then be further deconstructed into subtasks including “move the pen tip north,”

“move the pen tip east,” “move the pen tip south,” and, ”move the pen tip west.” Each listed event counted toward a student’s accumulation total.

Quantitative Methods Regarding Research Question 1b: Video Game Synthesis

A quantitative methodology was appropriate for examining Research Question 1b which focused on the mathematics of video game synthesis. Data from assessment of student game design journals employed during the synthesis phase of each game project were scored quantitatively to obtain measures of student content knowledge in mathematics during the synthesis phase of each video game. The specific quantitative measures were representations scores (Miles & Huberman, 1994; Glesne, 2006).

Representations scores. Following the recording of event recognition lists by treatment group students, the class engaged in a group discussion to complete a revised events list for the model video game. Students were then asked to synthesize each event on this revised events list using mathematical (i.e., numbers, symbols and equations) and computer programming (i.e., code, pseudocode) representations. A scoring matrix (Miles

& Huberman, 1994; Glesne, 2006) rating accuracy and completeness was developed to quantify student representations of their listed events. See the “Data Analysis” section for additional details on the scoring matrix.

Quantitative Methods Regarding Research Question 1c: Video Game Programming

Quantitative data was collected during the construction (programming) phase of each game project. Data consisted of tallies associated with elements of the completed video games. Tallies were collected to document the number of (a) required game events

69 included in each video game, and (b) mathematical modifications, if any, added to each game, and (c) aesthetic modifications incorporated in each game.

Qualitative Methods Regarding Research Question 1: Design and Construction

Qualitative data employed during the entire cycle of design and construction (i.e., analysis, synthesis, and programming) for each game project was also collected. Artifacts included (a) initial and final self-reflections documented in student game design journals,

(b) observations, (c) talk-alouds, and (d) completed game files (Kafai, 1995).

Self-reflections. Prior to and following activities associated with the construction of each project, students were asked to reflect on their plans and progress in producing their own video games. Design journal pages included prompts to obtain student reflections on successes and challenges. Design journals also prompted students to address their personalization of games by reflecting on their modifications and deviations from the theme and graphics of the model cover story (i.e., the game characters, setting, and mission) and game play (i.e., events) provided by the researcher. The creative process of personalization was not only fundamental to the development of new video games, but frequently rooted in the transfer of mathematical content knowledge from one context to another. For example, during the construction of the Frogger video game, a student may have decided to construct interlacing lanes of traffic (in lieu of the standard parallel lanes), thereby bringing knowledge to bear about perpendicularity in the game playfield. In contrast, altering only the characters and background scenery with no change to the underlying mathematics and programming did not constitute a transfer of mathematical content. Student reflections about these adaptations provided insight about both the

70 mathematical and aesthetic transfer students employed in the production of fully functional video games.

Completed video game files. Video game files in MicroWorlds EX were saved at the end of each class and at the end of each completed project. Each computer file was saved with a filename indicating the date and each student’s unique identification.

Mathematical achievement exhibited by students constructing functional video games in

MicroWorlds EX was corroborated by student reflections in their game journals.

Qualitative relationships between perceived progress (as measured by student reflections on successes and challenges) and actual progress in game construction were sought.

Additionally, trends in modifying the researcher-provided cover story and events in the video game model were examined.

Observations. In this study, qualitative data also included observation, “a fundamental and highly important method in all qualitative inquiry” (Marshall &

Rossman, 2006, p. 99). As the classroom teacher of the math enrichment course, and thus the direct provider of treatment instruction, the researcher took part in observation activities during each meeting session as a full participant. Immersion in the computer lab setting in which students engaged in video game instruction “permit[ted] the researcher to hear, to see, and to begin to experience reality as the participants do” (Marshall &

Rossman, 2006, p. 100).

Observations of students, the work they conducted and their interpersonal interactions with peers and the researcher were made during all treatment meetings.

Observations during the treatment phase addressed, “broad areas of interest but without predetermined categories or strict observational checklists… to discover the recurring

71 patterns of behavior and relationships” (Marshall & Rossman, 2006, p. 99). Four strategies guided observations, namely that they were conducted, via broad sweep; of nothing in particular; searching for paradoxes; and searching for problems facing the group (Wolcott,

1981). Observational field notes and researcher-teacher head-notes (Glesne, 2006) were written immediately following the conclusion of each class.

Talk-alouds. Following the completion of each video game project, the researcher invited individual students to participate in talk-alouds, discussing their game design and construction. Talk-alouds preserved the temporal properties of the cognitive process and offered insight into the steps used by each student in executing the process (Ericsson &

Simon, 1993). Recording and transcriptions of these talk-alouds, in conjunction with student reflections in the game design journals and saved student game files, provided further qualitative illumination of quantitative data.

Quantitative Methods Regarding Research Question 2: Transfer of Math Content

Quantitative methods were appropriate for describing (a) initial differences between study groups, (b) participant achievement in mathematics content knowledge, (c) achievement comparisons of groups on mathematics content tests, and (d) attitude comparisons on attitude inventory scales.

Standardized tests. Quantitative measures of baseline mathematical abilities (e.g., previously administered standardized math test scores) were used to identify participants to populate the study. Specifically, students with average scores of 85% or higher were admitted to the study population (i.e., they were all honors math or skip-grade math students). Then, students who also enrolled in math enrichment became the treatment group. The remaining high-performing math students became the comparison group.

72 Limitations regarding the school environment (e.g., scheduling), did not allow for randomization of treatment and comparison participants. To compensate for this, the researcher “attempt[ed] to ensure fair comparisons through means other than randomization, such as by using statistical techniques to adjust for background variables that may account for differences in the outcome” (National Research Council, 2002, p.

113).

At the outset of the study, statistical tests were performed to determine group normality and initial differences between the treatment and comparison groups. If initial differences in baseline mathematics performance or attitude existed, results would not be easily extensible to broader audiences. For baseline mathematics performance, difference tests were performed on achievement test scores obtained from the computational mathematics and quantitative reasoning sections of the norm-referenced Educational

Records Bureau (ERB, 2011) Comprehensive Testing Program (CTP) tests administered during late Spring 2008. Difference tests were also executed on researcher-developed, studywide mathematics content tests. For attitude performance, difference tests were performed on the pre-treatment administration of the Attitude Towards Mathematics

Inventory (ATMI) for both groups. See “Data Analysis” for additional details on statistical analysis of the tests of initial differences.

Studywide mathematics content tests. Quantitative measures of mathematics content knowledge, obtained via studywide pre-treatment and post-treatment mathematics content tests, were obtained for treatment and comparison groups. The use of quantitative methods to examine mathematics skill exhibited through tests is supported by previous studies examining the impact of Logo software and its derivatives (Klahr & Carver, 1988;

73 De Corte, 1992; Littlefield el al., 1988) and the specific methods through which instructional strategies are most effective (Kafai, 1995; Clements & Sarama, 1993).

Because this study sought to investigate the transfer of mathematics skills acquired from video game design and construction to traditional mathematics achievement tests, the research tested a correlational hypothesis: that participants programming video games in

MicroWorlds EX would exhibit higher performance levels on a multiple-choice, standards-based mathematics content test than participants not designing video games.

The studywide mathematics content test was developed by the researcher and consisted of multiple-choice questions addressing NCTM content standards reflecting with content required to construct the three video games.

Checkpoint mathematics content tests. Within the treatment group, student progress in mathematical knowledge was assessed at incremental stages between the start and end of the treatment period. As treatment group students were engaged in three video game projects, each project constituted a distinct stage at which mathematical content knowledge relevant to the specific project could be measured. To this end, game checkpoint pre-project and post-project content tests of mathematics were administered.

Game checkpoint tests consisted of problems similar to subsets of the studywide mathematics content test (i.e., those content areas addressing the content required to construct the game.

Quantitative and Qualitative Methods Regarding Research Question 3: Attitudes

An effective method of gauging student attitude is to conduct an attitude inventory.

In addition to measuring content knowledge changes associated with student design of video games, this study sought to profile students’ affective changes towards mathematics

74 associated with their design of video games. To this end, a quantitative methodology was appropriate for measuring and comparing, via Likert scales, the affective disposition of participants (Dutton and Blum, 1968; Shrigley and Koballa, 1987; Machi and Depool,

2002).

Attitude scores. Existing attitude inventories relevant to mathematics were examined for possible use. Specifically, an inventory was sought that elicited the types of information this study intended to discover. The Fennema-Sherman Mathematics Attitude

Scales (Tapia & Marsh, 2004) have been used extensively for measuring math attitudes, but focused heavily on gender-related themes. Since gender was not a research question addressed in the proposed study, it was decided that the Fennema-Sherman test was not appropriate in this venue. Other surveys examined did not possess age-appropriate phrasing or questions relevant to the experiences of middle school students, male and female.

It was ultimately decided that the Attitudes Toward Mathematics Instrument

(ATMI) instrument developed by Tapia and Marsh (2000) was the best choice for use with the study population of the proposed research (See “Instruments”). The ATMI measures numerous dimensions of student attitudes and have been proven valid through extensive field-testing. Thus, the ATMI was administered to all study participants before and after treatment activities. Statistical tests were executed to compare pre-treatment and post- treatment attitudes within and between groups, and to find relationships between affective dispositions and cognitive mathematics abilities as demonstrated on studywide content pre-tests and post-tests.

75 Observations and talk-alouds. Observational notes and notes transcribed during individual student talk-alouds as well as class discussions conducted by the researcher also provide clarification of student attitudes towards mathematics and video game design and construction. The notes were examined for trends within the group and for individuals in the context of activities undertaken during treatment.

Study Participants

Study participants consisted of 6th and 7th grade students at a private school in Las

Vegas, Nevada. Students ranged in age from 11 to 13 years old.

Treatment Group

The treatment group consisted of 19 students enrolled in an elective math enrichment course. These students were high performing in math, scoring in the 90th percentile or better on grade-level achievement tests in computational mathematics and quantitative reasoning. They represented the top 35% of their grade levels and consisted of ethnically diverse boys and girls. While only high-performing students were eligible to enroll in math enrichment, (approximately 50 students total), only a subset of those eligible chose to take the elective course. Mitigating factors, including scheduling conflicts, impacted whether students who registered for math enrichment actually enrolled in the course. Additionally, room capacity limited maximum course enrollment to 20 students. Appendix A presents example profiles of two students in the treatment group.

Comparison Group

The comparison group consisted of 6th and 7th grade students at the participating school who were high-performing in mathematics (i.e., taking 6th or 7th grade honors

76 math courses) but not enrolled in math enrichment. This high-performing subset of the general 6th and 7th grade population provided a more equivalent basis for comparison with the treatment group than the remaining, general population of 6th and 7th graders who were not enrolled in the math enrichment course. All potential comparison group students were invited to take part in the proposed study and all consented to do so.

Twenty-four students comprised the comparison group.

Non-Random Assignment and Initial Differences

Achievement test scores in mathematics and quantitative reasoning for the study population were obtained to provide a quantitative measurement of pre-treatment student math abilities. Because treatment group and comparison group students were not selected via random assignment from the larger student population of 6th and 7th graders (or even the high-performing subset of 6th and 7th graders), it was necessary to acknowledge initial differences in their baseline mathematics abilities.

Setting

The setting for this investigation was a small, independent day school providing instruction to approximately 700 students in grades pre-kindergarten through eight.

Located in Las Vegas, Nevada, the school is one of a small number of non-parochial private schools offering an alternative to public school education provided by the Clark

County School District, the nation’s fifth largest school district. The school admits an ethnically diverse student population with students possessing above-average academic abilities and few major learning or physical disabilities. Paying an average tuition of more than $20,000 annually, parents are of high socio-economic status, including business

77 owners and professionals. No student is classified as an English language learner (ELL) although some students speak multiple languages including Spanish, French, and

Mandarin. Learning experiences balance collaboration with competitiveness and include academic disciplines, fine and performing arts, STEM (science, technology, engineering and mathematics) and health and fitness at all grade levels. Cross-grade collaboration is encouraged and supported. Teachers are required to possess a minimum of a master’s degree and 3 years experience before teaching at the school. Many have taught at similar independent schools around the world. With these parameters in mind, careful consideration must be given to the generalizability of results to broader populations.

Regarding the physical environment, the study required the use of a school-based computer lab (Windows PCs were used) with the MicroWorlds EX program installed on each computer. Researcher access to student files via the school network was required to obtain copies of student work for future examination. Additional tables and chairs in the lab served as the seating area for students writing in their design journals and taking mathematics content tests and attitude inventories.

Many students in the study population have previously used the MicroWorlds EX software. Informal pilot studies addressing video game construction with students who had subsequently graduated (McCue, 2009) shaped the development of the current research. Most students participating in the treatment group and some students in the comparison group possessed experience using MicroWorlds EX in more traditional Logo contexts, primarily constructing geometric figures and determining relationships between polygons and turtle turn angles. However, study participants had not engaged in video game design and programming activities during previous school semesters.

78 Data Collection

A hybrid descriptive and quasi-experimental design was used to investigate the relationship between constructing video games and mathematics content knowledge.

Quasi-experimental parallels true experimental design in this case with the exception that there was no random assignment of groups. One challenge in conducting research in this format was the inability to shield the student participants from their status as either a treatment or comparison group participant. Nonetheless, the study attempted to conduct experimental practice, controlling for extraneous variables to the greatest extent possible

(Gay & Airasian, 2003). In this way, results may be more likely generalizable to settings beyond the current context.

Sequence of Study Activities

The sequence of study activities was executed during a single school year and is summarized in Figure 2. The study activities commenced in early November 2008 and concluded in late May 2009.

Digital toy Video game Video game Treatment Studywide project 1: project 2: project 3: Studywide Group math pre- analysis, analysis, analysis, math post-

test and synthesis, synthesis, synthesis, test and

attitude construction construction construction attitude

inventory inventory Control

Group

Figure 2. Sequence of study activities.

79 Outset, data collection. At the outset of the study, all participants (i.e., both treatment and comparison groups) engaged in a studywide pre-test of mathematical content knowledge. All participants also took a studywide pre-treatment attitude inventory that measured attitude towards mathematics.

Treatment period, general overview. During a 7-month period, the treatment group underwent a longitudinal intervention in which three video games were constructed.

For treatment group students, the intervention transpired in addition to their regularly scheduled, daily mathematics course. The comparison group did not engage in any video game programming nor special mathematics instruction beyond their regularly scheduled, daily mathematics course.

The treatment phase was conducted in three segments with one video game project per segment. The actual time available, in terms of number of 45-minute class periods, for each project was extremely flexible, with Etch-a-Sketch occupying approximately eight class periods, Frogger occupying approximately 12 class periods, and Tamagotchi Virtual

Pet occupying approximately 17 class periods. This flexibility allowed as much time as was needed for students to complete their checkpoint tests, initial and revised event lists, representations, design journal reflections and plans, and game programming.

For each game project, students in the treatment group followed a specific cycle of activities: (a) checkpoint mathematics pre-test, (b) game analysis, including interaction with the game model and listing events, (c) game synthesis, consisting of event representation, (d) game construction (writing of Initial Reflections and Design Plans in the design journals, programming in MicroWorlds EX, writing of Final Reflections in the design journals, and game talk-alouds), and (e) checkpoint mathematics post-test.

80 Video game and digital toy models created by the researcher served as the basis for each project. Projects analyzed, synthesized and constructed consisted of Etch-a-Sketch,

Frogger, and Tamagotchi Virtual Pet. The mathematics content encompassed by each game follows.

1. The Etch-a-Sketch digital toy addressed boundary values, inequality statements, heading, and addition/subtraction as inverse operations.

2. The Frogger video game required the use of Cartesian coordinates, the equation of a line and the speed equation.

3. The Tamagotchi Virtual Pet video game addressed variable expressions, variable manipulation and simple probability.

These three projects were selected because they represented mathematics concepts that were both age-appropriate (NCTM Standards, 2000) and relatively easy to translate to programming code to construct finished games.

Game production initially required some direct instruction to teach programming basics including vocabulary, syntax, and logic. The focus however, was not on programming itself, but on programming code as a mathematical representation and a route to mathematical problem solving. Milbrandt (1995) stressed the importance of contextualizing the programming, noting that, “the programming language is of secondary importance, with emphasis placed on the problem to be solved and the logical steps required for its solution” (p. 27).

Treatment period, data collection, pre-analysis. Each game project cycle began with the administration of a short, checkpoint mathematics pre-test relevant to the game to

81 be addressed. The checkpoint tests consisted of seven to ten, multiple-choice mathematics content problems.

Treatment period, data collection, analysis. The analysis phase then began with the researcher showing the treatment group the first project model, namely Etch-a-Sketch, constructed in MicroWorlds EX. Students were able to play with the digital toy on their computers, but were not able to access the programming code. In this way, operation of the toy was demonstrated without revealing any underlying mathematics or programming code.

Treatment group students then began an analysis of play in which they were first asked to identify toy elements and operation during a group discussion. Students were asked to work individually as they examined the toy for all events defining the game action. Definitions and simple examples of events were provided by the researcher in order to model the expected format for the students. Students then wrote their list of events in their game design journals. A simple count of events was then tallied to quantify students’ initial ability to define game play.

Because students frequently missed listing events in the video game, further identification required to fully describe game play was prompted by the researcher in a class discussion. Following class consensus on the revised, comprehensive event list, the researcher provided each student with a copy of the revised events list for their game design journals.

Treatment period, data collection, synthesis. The revised, comprehensive event list was formatted in three columns: the first column was the list of events, serving as the verbal descriptive representation; the second and third columns were blank. During

82 synthesis, students wrote their event representations, using the second column for mathematical representations, and the third column for programming code representations, if they knew them. Representations were scored on a matrix that rated the accuracy and completeness of student responses. These scores further quantified mathematical understanding during the synthesis phase.

Treatment period, lessons learned from pilot study. The processes of event analysis and synthesis were previously pilot tested with a small group of students (in fifth and eighth grades) similar to those of the study treatment group. Pilot students were shown a game, specifically Frogger, and asked to list four events that they could identify in the game. Students were then asked to provide verbal descriptive, mathematical and programming code representations of their events, given relevant examples.

While listing events in the pilot study, some students chose to write mathematical representations of events closely related to the example event. In 4C/ID terms, these events were tasks in the same task class (van Merriënboer’s & Kester’s 4C/ID, 2005).

These students produced correct or nearly correct mathematical representations and programming representations. Other students chose to describe more complex events that were well differentiated from the example. While these students were capable of writing mathematical representations, the accuracy of these representations varied. Additionally, without a translation model, only one student who was already somewhat familiar with programming in MicroWorlds EX was capable of translating his mathematical representation into code (with partial accuracy).

Treatment period, data collection, pre-programming. The construction phase then began, with the researcher inviting treatment group students to plan the appearance of

83 their games in MicroWorlds EX. Students recorded these initial design plans in their game design journals. The researcher informed them that they could deviate from the original design of the game model so long as they preserved the general structure of the game play

(i.e., no characters, goals or events were removed, only modified or amplified). Students wrote descriptions and drew graphics to show their own versions of the game that they intended to construct (first Etch-a-Sketch, second Frogger, third Tamagotchi Virtual Pet).

Students were also asked to note any new mathematical representations and programming commands required to implement their plans. They also reflected on their successes and challenges experienced in analyzing the game models and planning their own versions of the games. Initial design documents were used to corroborate quantitative data obtained from the checkpoint pre-test and the event lists and event representations.

Treatment period, data collection, programming. The researcher then led students into the programming phase of game construction. During this period, students used their previous analyses and syntheses recorded in their game journals to write programming code in a MicroWorlds EX game file. Students already recognized that game play was broken down into component task and subtask problems (i.e., events) which were more easily solved (i.e., van Merriënboer’s and Kester’s 4C/ID model for the instruction of complex tasks, 2005). The tasks and subtasks were then ultimately reassembled to construct the whole game. Worked examples, guided discussion, and peer- to-peer interaction provided models and support to students in their programming efforts.

At the end of each treatment meeting conducted during game construction, students were asked to save their game files with a name and date stamp. Following

84 completion of the fully finished game file, students saved their games as “FINAL,” along with their names in the filename.

As students shared their games and invited one another to play their finished products, the researcher circulated, making observations and recording candid talk-alouds to obtain oral reflections by the treatment group students of their work.

Treatment period, data collection, post-programming. Students were asked to create a Final Reflections page in their game design journals, reflecting on their progress in programming game events into a functional video game, and noting successes and challenges they encountered. Students were also asked to sketch their completed games as well as describe modifications and deviations they made from the original game model.

Post-programming activities closed with the administration of the checkpoint mathematics post-test (the same instrument as the pre-test) addressing content relevant to the video game just constructed.

Close of study, data collection. Following the completion of the treatment phase, all study participants took a studywide post-test of mathematical content (the same instrument as the pre-test) and completed a post-treatment inventory of mathematics attitude (the same instrument as the pre-treatment inventory).

Study Timeline

Research activities were conducted during the Fall and Spring semesters (October

2008 to May 2009) of the 2008-2009 academic year.

Three additional meetings (each 45 minutes in duration), were required of all study group participants to complete the consent process, administer the studywide pre-tests and

85 post-tests of mathematics content, and administer the pre- and post-treatment attitude inventories measuring student disposition toward mathematics.

Approximately 37 course meetings, each 45 minutes in duration, were conducted as treatment phase class meetings with students in the math enrichment course. During these meetings, students engaged in checkpoint pre-tests and post-tests and in video game project activities consisting of game analysis, game synthesis, and game construction.

Instruments

A range of instruments were employed throughout the study to collect data regarding student performance in mathematics, programming, and attitude. Descriptions of specific instruments and their usage in the study follow.

Instruments, Studywide Mathematics Content Tests (Pre/Post Study)

A test of mathematics content (see Appendix B) was administered to all participants twice during the study, specifically at the commencement of the study as a pre-test and at the termination of the study as a post-test. The test was constructed by the researcher and consisted of 20 multiple-choice questions addressing concepts and processes prescribed by the National Council of Teachers of Mathematics standards

(NCTM, 2000). Specific questions were written to reflect the types of mathematical content and processes addressed through video game interventions during the treatment phase of the study. Questions did not directly address programming syntax nor

MicroWorlds EX code. Students were allowed 30 minutes to complete the test at each administration.

86 Three questions (i.e., problems) on the mathematics content test are discussed to demonstrate the types of problems students were asked to solve. Test item one (see Figure

3) addressed the “Use coordinate geometry” item of the geometry standard for grades 6-8

(NCTM 2000). Game design activities in Frogger addressed content relevant to this type of question among students in the treatment group. Item eight (see Figure 4) encompassed the “Write inequalities to describe boundaries” item of the algebra standard for grades 9-

12 (NCTM 2000). Game design activities featured in the construction of the Etch-a-Sketch addressed this type of question. Test item eighteen (Figure 5) addressed the “Write

Algebraic expressions to manipulate variables” item of the algebra and measurement standards for grades 6-8 (NCTM 2000). Treatment group students building Tamagotchi

Virtual Pet explored content relevant to this question type.

Figure 4. Studywide mathematics content test item addressing coordinate geometry.

Figure 5. Studywide mathematics content test item addressing inequality graphing.

Figure 5. Studywide mathematics content test item addressing variable manipulation.

Because there were no existing mathematics content tests comprised of the precise mix of standards-based questions relevant to the video games addressed in the treatment phase, the researcher developed a studywide mathematics content test specifically for this

88 purpose. Questions were based on NCTM standards (2000) with approximately one-third of the test addressing questions derived from Etch-a-Sketch, one-third from Frogger and one-third from Tamagotchi Virtual Pet. Additionally, questions were written in multiple- choice format and in similar form to questions presented by NCTM for classroom use.

Most questions on the studywide mathematics content test addressed NCTM standards for grades 6-8. However, because Algebra I is a course now taught in the middle grades – and a course taken by 7th graders who are high-performing in mathematics as well as all eighth graders, questions relevant to Algebra I themes were included on the mathematics content test. NCTM categorizes these Algebra I standards in the grades 9-12 group. Appendix B consists of a complete copy of the studywide mathematics content test.

Instruments, Checkpoint Math Content Tests (Pre/Post Video Game Project)

Checkpoint tests were administered within the treatment group before and after each video game project. Checkpoint tests consisted of questions similar in format to those on the studywide mathematics content test. However, only those types of questions relevant to each of the three video games were included in the checkpoint tests administered in conjunction with that game. Students were given 15 minutes to complete each checkpoint test.

Instruments, Student Game Design Journals

Replicating a successful tool employed in Kafai’s video game design research

(1995), treatment students maintained game design journals during game analysis, synthesis, and construction activities. Game design journals were folders consisting of pages on which students were asked to record specific information about their game design and construction activities.

89 Information recorded by students in their game design journals consisted of the following.

1. During game analysis, an initial list of events defining game play in the game model.

2. During game synthesis, a table of representations describing each event, written in verbal (i.e., the revised event list), mathematical, and programming code forms.

3. At the beginning of game construction, a reflections and plans document (i.e.,

(initial reflections and design plans) that included a student’s perception of his or her successes and challenges in writing events and creating event representations; a labeled sketch depicting the game environment (background and characters) that the student planned to create in MicroWorlds EX; and a discussion of any modification the student intended to implement from the game model.

4. At the end of game construction, a reflections and plans document (i.e., Final

Reflections) that included a student’s perception of his or her successes and challenges translating written programming code to a functional game file in MicroWorlds EX on the computer; a labeled sketch depicting the completed game environment (background and characters) that the student created in MicroWorlds EX; and a discussion of any modifications the student implemented from the game model.

Instruments, Video Game Files

At the end of each treatment meeting conducted during video game construction, students saved their work-in-progress game file. Students were asked to save in a format that included the video game tag, student initials and the date (e.g., EtchWW915 for “Etch

Will Wright November 15”). Students were instructed not to overwrite older versions of

90 game files composed during previous meeting dates, although they frequently did so accidentally. Ultimately, the final, saved video game for each student served as the artifact for corroborating other quantitative and qualitative data.

Instruments, Attitudes Inventory

Student attitudes towards mathematics in general and as a function of participation in the video game design study were measured for study participants. Understanding the role of student disposition, including attitudes, motivation and other affective facets, is important in shaping learning activities in which students are truly involved.

Disposition of all study participants was measured via a pre-treatment survey at the commencement of the study and again via a post-treatment survey (the same instrument) at the end of the study. Several validated instruments for measuring disposition towards mathematics were examined, including the popular Fennema-Sherman Mathematics

Attitude Scales (Tapia & Marsh, 2004). However, the Attitude Toward Mathematics

Inventory (ATMI) survey instrument was found to be superior for measuring student disposition in this study (see Appendix C). Not only has ATMI been proven valid through repeated research (Tapia & Marsh, 2004), it has also been validated with an audience nearly identical to the study population – high socioeconomic, middle school students

(Tapia & Marsh, 2000).

The ATMI consisted of 40 items in four categories, addressing (a) confidence, (b) value, (c) enjoyment, and (d) motivation. Response choices were formatted on a 5-point

Likert scale, consisting of 1: strongly disagree; 2: disagree; 3: neutral; 4: agree; and 5: strongly agree. Several items were reversed and were given the appropriate value for data analysis (Tapia & Marsh, 2000).

91 Internal consistency of the ATMI instrument as measured by Cronbach alpha is

.95, the split-half reliability is .83 and the Spearman-Brown reliability is .91.

Additionally, the ATMI instrument mean is 144.54, standard deviation is 24.99 and the standard error of measurement is 5.42. All items possess item-to-total correlations of .45 or greater, meaning that all items contribute significantly to their categories. Thus, the test items are homogenous, measuring a single, common trait within each category (Tapia &

Marsh, 2000).

Mitigating Potential Threats to Validity

Measurement of student achievement in technology-infused educational settings has been a critical research focus for several decades. Since the purpose of school-based research is to inform education within the school and ultimately to extend findings to the broader educational community, researcher-teachers must anticipate potential threats to the validity of results.

Threats to Validity, Role of the Researcher

Researcher-teachers inevitably influence the environment and interactions they investigate, with the act of observing affecting the phenomenon being observed (Merriam,

1998). The researcher in this study was the enriched mathematics teacher of most 6th grade treatment participants for 2 years; the enriched mathematics teacher of most 7th grade treatment participants for 3 years; and the technology teacher of some 7th grade treatment participants for 1 year. The researcher had previously conducted technology research addressing gender-based preferences of video games (McCue, 2008) with several students in the treatment group. Thus, there was a comfort level and familiarity already established between the researcher and the treatment group.

92 The role of the teacher in fostering educational success is intricately bound up with other facets of the classroom environment. The National Mathematics Advisory Panel of the United States Department of Education, in its recently released Foundations For

Success: The Final Report (2008) document, executed a meta-analysis on current mathematics research to compile a list of findings and recommendations for instruction and future research. The Report stated that, “The Panel recommends that computer programming be considered as an effective tool…, for developing specific mathematics concepts and applications, and mathematical problem solving abilities. Effects are larger if the computer programming language is designed for learning (e.g., Logo) and if students’ programming is carefully guided by teachers so as to explicitly teach students to achieve specific mathematical goals (US Dept. of Ed, 2008, p. 52).” Moursund (1997) reinforced this viewpoint, pointing out that, “Logo and the environments created from it does not automatically guarantee educational success. While IT-assisted PBL [instructional technology-assisted project based learning] is an excellent vehicle for implementing a constructivist theory of teaching and learning a significant contributor to student success is the teacher’s knowledge and skills” (p. 36). Thus, the role of the classroom teacher in providing instruction and motivational support to the treatment group students must be carefully considered. During the course of the study, the researcher-teacher maintained an active participant role in treatment activities, interacting directly with the students under investigation (Spradley, 1980). However, because of the constructionist nature of video game design, the researcher engaged in less of a “direct instructor” role than most traditional teachers. This de-emphasis of the didactic allowed the researcher to step back more frequently into the role of observer, taking note of overall student experiences in the

93 game construction process. Care was given to avoid inadvertently skewing or contaminating outcomes in the treatment group. Ultimately, though, the influence of the researcher-teacher was not easily dissected from the instructional intervention.

Threats to Validity, Sample Selection

Another potential threat to validity was the problem of selecting samples representative of the school population at the study site. Because treatment and comparison group students were the highest-performing mathematics students at the school, results achieved from studying these groups were not easily extensible to the greater student body. Beyond the school population, results may not be extensible to students of similar abilities in other independent-school environments. Generalizations to typical populations in public school settings may be tenuous at best.

Threats to Validity, Instruments

A key measurement instrument, specifically the studywide mathematics content test, posed a threat to validity. As the content test was written by the researcher and was not pilot-tested nor proven valid, its use presented some difficulty in proving that it measured what it was intended to measure.

Threats to Validity, Maturation and Morbidity

Participant maturation may present a threat to validity as well. Maturation – changes arising from the natural physical, emotional, and intellectual evolution of subjects over time – is of special relevance to the 6th and 7th grade audience under consideration.

Because young teens experience significant developmental changes (Rice & Dolgin, 2002;

Slavin, 2000), qualitative observations must attempt to discern whether treatment effects result from treatment interventions and not personal growth. In this study, maturation also

94 may have resulted from content knowledge development resulting from out-of-class practice with the relevant mathematics during their regularly scheduled mathematics classes; or with the MicroWorlds EX software, either at home or during unassigned time spent in the school computer lab.

Lastly, mortality, the circumstance in which students leave the study, may have also presented a threat to validity. During this research, no study participant exited the study. One student left the school during the last month of the school year, but his mother delivered and administered the post-treatment attitude survey (the only instrument he had yet to complete) to him and then returned the completed survey to the researcher.

Factors that might have typically threatened validity that did not require consideration in this study include access to the computer lab facility and the presence of fundamental technology skills among participants. The researcher was permanently assigned to the computer lab full-time, during all scheduled classes. Further, students engaged in weekly technology classes beginning in Kindergarten, with most core teachers including additional technology activities in academic lessons.

Data Analysis of Quantitative Measures

Descriptive and inferential statistics were used to answer the four research questions. Descriptive statistics including mean and standard deviation (SD) were computed to report performance on the following measures.

1. Studywide mathematics content tests.

2. Checkpoint mathematics tests.

3. Tally of initial game play events recorded by students in design journals.

95 4. Ratings of event representations recorded by students in design journals.

5. Likert-scale attitude inventory responses.

Statistical analyses were carried out using the Statistical Product and Service

Solutions (SPSS) software. Means and standard deviations (SDs) were computed for both preperformance and postperformance on tests and inventories by group. Means and standard deviations were also computed for tallies of listed game play events and matrix ratings of event representations. Statistical tests were used to explore relationships within and between groups as described below. Additionally, the statistical significance of variables relevant to each research question was computed in SPSS.

Valdez (2004) noted that, for statistical results to prove useful, “most educational researchers, especially those who have examined large numbers of studies (meta- analyses), agree that if used appropriately, technology can improve education in the effect- size range of between 0.30 and 0.40.” Effect size is a numerical way of expressing the strength or magnitude of a reported relationship. Cohen (1977) extended this benchmark by providing ranges of effect-size utility, classifying effect sizes of around 0.2 as small,

0.5 as moderate and 0.8 as large. The moderately small sample sizes of N=19 for the treatment group and N=24 for the comparison group limit the power rating (level of significance) to .4 with an effect size of .54 (Lenth, 2001).

Data Analysis of Research Question 1: Event Tallies, Representations, Event

Inclusions, and Game Modifications

Within the treatment group, students initiated game analysis by exploring the operation of a video game model created by the researcher in MicroWorlds EX. Students

96 in the treatment group then developed a list of events defining game play. A simple count of events was tallied.

Following group discussion led by the researcher, students were given a revised event list that precisely defined game play for the game model. The events in this revised list became the verbal representations.

Students then worked to represent each event using mathematical notation and programming code. Representations were scored using scoring matrices which employed

“+”, “0”, and “–“ codes. Negative (“–1“) codes were scored for representations not attempted or substantially incorrect. Neutral (“0“) codes were scored for attempts that included only partially correct or incomplete representations. Positive (“+1“) codes were recorded for representations which were mostly complete and which contained mostly correct statements describing the event (Miles & Huberman, 1994; Glesne, 2006).

Patterns of performance exhibited by treatment group students as they progressed through the analysis phase from the first project, Etch-a-Sketch, to the second project,

Frogger, to the third project, Tamagotchi Virtual Pet, were sought. It was posited that, during game analysis, students would progress from generating less accurate representations on Etch-a-Sketch towards more accurate representations in later games.

This shift would be scored as a move from lower scores on representation ratings in Etch- a-Sketch to higher scores on representations in Frogger and Tamagotchi. Further, a shift towards more accurate ratings would imply that, as they gained experience, students were becoming more successful in understanding all necessary mathematics and programming elements required to bring a functional game to fruition. Mathematically, representation ratings could range from very negative scores, demonstrating a low level of

97 understanding, to very positive scores, demonstrating a high level of understanding. The composite score of a student’s entire set of representations equaled the sum of per-event scores for all events in a given video game.

Once treatment students completed constructing their games, a tally was made for each student’s completed game to determine how many events from the game model were included. Finally, a tally of modifications created by each student in their completed versions of Etch-a-Sketch, Frogger, and Tamagotchi Virtual Pet was performed.

Data Analysis of Research Question 2: Lateral Transfer of Math Ability

Lateral transfer of mathematics ability from the programming context to the traditional context of multiple-choice tests was conducted in two ways. First, lateral transfer was examined for treatment group students by project, for each of the three projects, namely Etch-a-Sketch, Frogger, and Tamagotchi Virtual Pet. Second, lateral transfer was examined for all study participants at the outset and at the close of the study, for the purpose of comparing performance between treatment and comparison groups.

Analysis of lateral transfer of math within the treatment group. Prior to and following each video game construction project, treatment group students completed a short checkpoint test of mathematics content relevant to the current game. Each checkpoint test consisted of approximately ten multiple-choice problems. Because treatment group participants were math enrichment students, samples were not assumed to be normally distributed. Therefore, simple descriptive and nonparametric statistics of checkpoint test scores pre- and post-game were employed to examine results.

To determine within group shifts in student performance associated with the construction of each game, the pre-test to post-test shift for responses to each question

98 item was measured (Roy, 2001, Draugalis & Jackson, 2004; I-TECH, 2008). Three answer possibilities existed for most questions on the checkpoint content tests: correct (C), acceptable (Acc), or incorrect (I). An acceptable response was one that indicated that the student had partial understanding of how to answer the question, but missed a detail in reaching the completely correct response. Some checkpoint content questions consisted of only a correct response (C) or an incorrect (I) response.

In scoring the checkpoint content test, the shift by a student from an item response on the pre-test to an item response on the post-test was measured as follows.

1. Positive change (a shift towards a more correct response).

2. No loss (an acceptable response followed by another acceptable response, or a correct response followed by another correct response or an acceptable response).

3. No gain (an incorrect response followed by another incorrect response).

4. Negative change (a shift towards an incorrect response).

Percentages of response shifts measured as positive change, no loss, no gain, and negative change were then reported. The treatment group sample size of N=19 limited the power rating and effect size statistics computed for checkpoint tests.

Analysis of lateral transfer of math between groups. Opening and closing the entire study, all study participants completed a 20-item multiple-choice test of mathematics content relevant to the study content. Because study participants were selected from honors and math enrichment populations, samples were not assumed to be normally distributed. Further, samples between comparison and treatment groups were unpaired and independent. Therefore, simple descriptive and nonparametric statistics of studywide test scores pre- and post-treatment were employed to examine results.

99 To determine studywide between group shifts in student performance associated with the treatment intervention, the pre-test to post-test shift for responses to each question item was measured (Roy, 2001, Draugalis & Jackson, 2004; I-TECH, 2008). For the studywide mathematics content test, student responses were scored as a correct (C) response, acceptable response (Acc), or an incorrect (I) response. An acceptable response was one that indicated that the student had partial understanding of how to answer the question, but missed a detail in reaching the completely correct response. Some checkpoint content questions consisted of only a correct response (C) or an incorrect (I) response. In computing the results of the studywide content test, the shift by a student from an item response on the pre-test to an item response on the post-test was measured as positive change (I  C), (I  Acc), or (Acc  C); no loss (C  C) or (Acc  Acc); no gain (I  I), or negative change (C  I), (C  Acc), or (Acc  I). Percentages of response shifts measured as positive change, no loss, no gain, and negative change from the pre-treatment to the post-treatment administrations of the studywide test were then reported. These results were then examined to determine between-group shifts in student performance associated with participation in the study.

Additional analysis was also performed by measuring and comparing the pre- treatment and post-treatment raw scores on the studywide content tests. To evaluate within group changes that occurred during the course of the study, raw scores earned by students in each of the treatment and the comparison groups on the studywide content tests were examined via a Wilcoxon Signed Ranks test. To evaluate between groups changes that occurred during the course of the study, raw scores earned by each group on the studywide content tests were examined via a Mann-Whitney U test. Both the Wilcoxon and the

100 Mann-Whitney U are nonparametric statistical tests (two-tailed), appropriate for examining the data, which was shown not to be normally distributed. The Wilcoxon

Signed Ranks test and the Mann-Whitney U test are analogous to parametric t-tests. A significance level of .05 was chosen for identifying whether within group changes or between group differences were significant.

Because members of the treatment and comparison groups were not randomly assigned, these nonequivalent groups required an “unequal variance” limiter be included in statistics comparing groups. The total sample size of N=43 students allowed for descriptive and inferential statistical procedures to be employed with acceptable power ratings and effect sizes in measuring and interpreting outcomes.

Data Analysis of Research Question 3: Attitudes Toward Mathematics

The ATMI attitude survey in which all study participants engaged at the beginning and end of the study period produced per-item ordinal scores of 1 through 5 for each of 40 statement items grouped into the four scales of confidence, value, enjoyment, and motivation. Means and standard deviations for each scale were computed for each of the treatment (N=19) and comparison (N=24) groups.

To evaluate within group changes that occurred in attitude towards mathematics during the course of the study, ATMI scores for students in each of the treatment and the comparison groups were examined via a Wilcoxon Signed Ranks test. To evaluate between groups changes that occurred in attitude towards mathematics, ATMI scores for the two groups were compared via a Mann-Whitney U test. Both the Wilcoxon and the

Mann-Whitney U are nonparametric statistical tests (two-tailed), appropriate for examining the data which was shown not to be normally distributed. The Wilcoxon

101 Signed Ranks test and the Mann-Whitney U test are analogous to parametric t-tests. A significance level of .05 was chosen for identifying whether within group changes or between group differences were significant.

Finally inferential statistical tests were employed in an attempt to extract relationships among student performance on mathematical content tests, performance in video game event analysis (event scores and representations ratings), and attitude towards mathematics as measured on the ATMI.

102 CHAPTER 4

RESULTS

This study evaluated the mathematics content learned and attitudes exhibited by students engaged in the design and construction of video games over several months.

During the study, treatment group students analyzed video games for their mathematical events; synthesized the mathematics of video game events; and programmed functional games.

Summary of Research Questions

Three research questions guided this study. The questions addressed learning mathematics content, transferring mathematics content knowledge, and mathematics attitude as follows.

Question 1, Mathematics of Video Game Design and Construction

This question entailed three parts: (a) analysis – What mathematics content do middle school students invoke as they analyze games? (b) synthesis – What mathematics do middle school students invoke as they synthesize games? (c) programming – What mathematics do middle school students invoke as they program games?

Question 2, Lateral Transfer of Mathematics Content Knowledge

This question entailed two parts: (a) On a standards-based, multiple-choice mathematics content test, how does the performance of middle school students change, pre- and post-design and construction of video games? (b) On a standards-based, multiple- choice mathematics content test, how does the performance of middle school students who are engaged in video game design and construction compare with the performance of

103 students of similar math abilities who are not engaged in video game design and construction?

Question 3, Attitudes Toward Mathematics

This question entailed two parts: (a) How can the attitude of middle school students towards mathematics be characterized prior to designing and constructing video games and after designing and constructing video games? (b) How do the attitudes towards mathematics compare between middle school students who are engaged in video game design and construction and those who are not engaged in video game design and construction?

To investigate these questions, 6th and 7th grade middle school students engaged in designing and constructing three video games using the MicroWorlds EX environment.

Treatment students designed and constructed (a) Etch-a-Sketch (b) Frogger and (c)

Tamagotchi Virtual Pet. They also took mathematics content tests and attitude inventories pre-treatment and post-treatment. Nineteen treatment group students and twenty-four comparison group students took part in the study.

Initial Differences Between Groups

At the outset of the study, the treatment and comparison groups were examined for normality using tests of skewness and kurtosis (Hopkins & Weeks, 1990). Tests were executed on (a) measures of mathematics performance on standardized tests, (b) measures of mathematics performance on studywide content pre-tests, and (c) measures of affective performance on pre-treatment attitude surveys. Criteria defined by Meyers, Gamst, &

Guarino (2006) were employed in determining whether measures exhibited normal

104 distributions. Following tests of normality, groups were compared to determine whether any initial differences existed.

Comparison by Achievement Test Scores

Groups were compared by performance on achievement test scores. This measurement was chosen because it provided a standardized basis of comparison for student mathematical performance, and because it existed for all study participants.

Achievement test scores on quantitative reasoning tests and on numerical tests were combined to form a composite mathematics achievement test score for each student in the study.

Achievement test scores, descriptive statistics. Composite mathematics achievement test scores are shown in Table 3. For the entire study group (N = 43), the mean composite mathematics achievement test score was 92.5 (SD = 6.0). For students in the treatment group (N = 19), the mean composite mathematics achievement test score was 96.1 (SD = 3.1). For students in the comparison group (N = 24), the mean composite mathematics achievement test score was 89.7 (SD = 6.3).

Table 3

Mathematics Achievement Test Scores by Group

Percentile Scores on Math Achievement Tests Group N Min Max Mean SD Treatment 19 86.5 99.0 96.1 3.1 Comparison 24 76.5 98.5 89.6 6.3 Combined 43 76.5 99.0 92.5 6.0

105 Achievement test scores, tests of normality. Statistical tests of skewness and kurtosis were conducted on the score data sets to examine whether composite mathematics achievement test scores earned by students in each of the treatment and comparison groups were normally distributed. See Table 4 for results of these normality tests.

For the treatment group, data points skewed negatively and the computed skewness statistic was -1.8 with a standard error of .52. Because the skewness statistic was more than twice its standard error, the achievement test scores of treatment group members showed a departure from normal symmetry. Achievement test scores of students in the treatment group showed a kurtosis of 4.3, more than four times its standard error of

1.0, revealing significantly more clustering than a normal distribution. Thus, on achievement test scores, the treatment group was not considered normally distributed.

The achievement test scores of the comparison group showed a skewness statistic of -.70 with a standard error of .47. Kurtosis was computed as -.32 with a standard error of .92. While the achievement test scores of the comparison group were skewed negatively and somewhat platykurtic, they did not depart significantly from normality. Since achievement test scores in the treatment group were not normally distributed, further statistical exploration of these measures was computed nonparametrically.

Table 4

Tests of Normality: Mathematics Achievement Test Scores by Group

Skewness Kurtosis Group Statistic Std. Error Statistic Std. Error Treatment -1.8 .52 4.3 1.0 Comparison -.70 .47 -.32 .92

106 Achievement test scores, differences between groups. To examine whether initial differences existed on achievement tests of the treatment and comparison groups, the Mann-Whitney U statistic was computed. The Mann-Whitney U is appropriate for nonparametric comparisons of performance between unpaired groups (Blaikie, 2003). On achievement test measures, the Mann-Whitney U was computed as 68.5 (p < .001) indicating that, at the outset of the study, treatment and comparison groups showed differences that were statistically significant.

Comparison by Studywide Content Pre-test Scores

Groups were compared by performance on a researcher-constructed, 20-item, multiple choice studywide mathematics content pre-test. Question items featured mathematical topics relevant to the mathematics addressed in the course of constructing video games in the study.

Studywide content pre-test scores, descriptive statistics. As shown in Table 5, the mean studywide content pre-test score for the treatment group (N = 19) was 10.2 out of 20 (SD = 2.7) and the mean studywide pre-test score for the comparison group (N = 24) was 9.0 out of 20 (SD = 1.9).

Table 5

Studywide Mathematics Content Pre-test Scores by Group

Group N Min Max Mean SD Treatment 19 5 15 10.2 2.7 Comparison 24 7 14 9.0 1.9 All participants 43 5 15 9.5 2.3

* Range of test was 0 (min) to 20 (max).

107 Studywide content pre-test scores, tests of normality. Statistical tests of skewness and kurtosis were conducted to examine whether studywide content pre-tests scored by each of the treatment and comparison groups were normally distributed. Results of these normality tests are shown in Table 6.

Table 6

Tests of Normality: Studywide Mathematics Content Pre-test Scores by Group

Skewness Kurtosis Group Statistic Std. Error Statistic Std. Error Treatment .09 .52 -.30 1.0 Comparison 1.1 .47 1.25 .92 All participants .63 .36 .10 .71

For the treatment group, the computed skewness statistic was .09 with a standard error of .52. Because the skewness statistic very close to 0, the studywide pre-test scores of treatment group members exhibited nearly normal symmetry. Studywide pre-test scores of students in the treatment group showed a kurtosis of -.30 with a standard error of 1.0, a mesokurtic distribution. Thus, on studywide pre-test scores, the treatment group was considered normally distributed.

The studywide pre-test scores of the comparison group showed a skewness statistic of 1.1. The skewness statistic was more than twice its standard error (.47) and thus the studywide pre-test scores of the comparison group did not exhibit normality. Kurtosis was computed as 1.3 with a standard error of .92. Therefore, the studywide pre-test scores of the comparison group were skewed positively and slightly leptokurtic, exhibiting a departure from normality.

108 The studywide pre-test scores of the entire study group showed a skewness statistic of 1.1. The skewness statistic was less than twice its standard error (.47). Kurtosis was computed as .10 with a standard error of .71. Therefore, the studywide pre-test scores of the group in its entirety exhibited normality.

In summary, studywide pre-test scores of the treatment group exhibited normality, while studywide pre-test scores of the comparison group were not normally distributed.

Studywide pre-test scores of the entire group exhibited normality Therefore, further statistical exploration comparing studywide test measures between these groups was computed nonparametrically.

Studywide content pre-test scores, differences between groups. To examine whether initial differences existed between studywide pre-tests of the treatment and comparison groups, the Mann-Whitney U statistic was computed. On studywide pre-test measures, the Mann-Whitney U was computed as 162.50. This value did not indicate statistical significance at the p < .05 level. Therefore, based on studywide pre-tests administered at the outset of the study, treatment and comparison groups showed no significant statistical differences.

Comparison by ATMI Affective Scale Scores

Groups were compared at the outset of the study by scores on each of the four scales of the Attitudes Toward Mathematics Inventory (ATMI, Tapia & Marsh, 2004) affective instrument: confidence, value, enjoyment, and motivation. Each scale consisted of statements requiring Likert-style responses on a five-point scale from participants, with several items reversed on each scale. ATMI scales, their respective number of statements, and their maximum values are shown in Table 7.

109 Table 7

ATMI Scales and Maximum Possible Values

Scale Statements Maximum Value Confidence 15 75 Value 10 50 Enjoyment 10 50 Motivation 5 25

Pre-treatment ATMI scores, descriptive statistics. Table 8 shows the results of the pre-treatment ATMI attitude inventory for both treatment and comparison groups. The mean pre-treatment ATMI confidence scale score for the treatment group (N = 19) was

66.0 out of 75 (SD = 5.7) and the mean pre-treatment ATMI confidence scale score for the comparison group (N = 24) was 58.5 out of 75 (SD = 8.0).

The mean pre-treatment ATMI value scale score for the treatment group (N = 19) was 47.3 out of 50 (SD = 3.0) and the mean pre-treatment ATMI value scale score for the comparison group (N = 24) was 43.6 out of 50 (SD = 4.3).

The mean pre-treatment ATMI enjoyment scale score for the treatment group

(N = 19) was 45.8 out of 50 (SD = 5.3) and the mean pre-treatment ATMI enjoyment scale score for the comparison group (N = 24) was 38.6 out of 50 (SD = 7.7).

The mean pre-treatment ATMI Motivation scale score for the treatment group

(N = 19) was 22.9 out of 25 (SD = 2.7) and the mean pre-treatment ATMI motivation pre- test scale score for the comparison group (N = 24) was 18.79 out of 25 (SD = 4.1).

110 Table 8

Pre-treatment ATMI Inventory Scores by Scale and by Group

Scale and group N Min Max Mean SD ATMI_Confidence – Treatment 19 47 / 75 70 / 75 66.0 / 75 5.7 ATMI_Confidence – Comparison 24 44 / 75 70 / 75 58.5 / 75 8.0 ATMI_Value – Treatment 19 39 / 50 50 / 50 47.3 / 50 3.0 ATMI_Value – Comparison 24 35 / 50 50 / 50 43.6 / 50 4.3 ATMI_Enjoyment – Treatment 19 32 / 50 50 / 50 45.8 / 50 5.3 ATMI_Enjoyment – Comparison 24 17 / 50 48 / 50 38.6 / 50 7.7 ATMI_Motivation – Treatment 19 17 / 25 25 / 25 22.9 / 25 2.7 ATMI_Motivation – Comparison 24 10 / 25 25 / 25 18.8 / 25 4.1

Pre-treatment ATMI scores, tests of normality. Statistical tests of skewness and kurtosis were conducted to examine whether ATMI scores for each scale were normally distributed for each of the treatment and comparison groups (see Table 9).

The pre-treatment ATMI confidence scores of the treatment group showed a skewness statistic of -2.4. The skewness statistic is more than twice its standard error (.52) and thus the pre-treatment ATMI confidence scores of the treatment group did not exhibit normality. Kurtosis was computed as 6.7 with a standard error of 1.0. Therefore, the pre- treatment ATMI confidence scores of the treatment group were skewed negatively and are leptokurtic, exhibiting a departure from normality.

The pre-treatment ATMI confidence scores of the comparison group showed a skewness statistic of -.06 with a standard error of .47. Kurtosis was computed as -1.1 with a standard error of .92. Therefore, the pre-treatment ATMI confidence pre-test scores of the comparison group exhibited almost no skewness and were only slightly platykurtic, indicating a distribution that was considered normal.

111 The pre-treatment ATMI value scores of the treatment group showed a skewness statistic of -1.1. The skewness statistic was more than twice its standard error (.52) and thus the pre-treatment ATMI value scores of the treatment group did not exhibit normality. Kurtosis was computed as 1.7 with a standard error of 1.0 Therefore, the pre- treatment ATMI value scores of the treatment group were skewed negatively and slightly leptokurtic, exhibiting a departure from normality.

The pre-treatment ATMI value scores of the comparison group showed a skewness statistic of -.48 with a standard error of .42. Kurtosis was computed as -.59 with a standard error of .92. Therefore, the studywide ATMI value scores of the comparison group exhibited little skewness and were mostly mesokurtic, indicating a distribution that was considered normal.

The pre-treatment ATMI enjoyment scale scores of the treatment group showed a skewness statistic of -1.9. The skewness statistic was more than twice its standard error

(.52) and thus the pre-treatment ATMI enjoyment scores of the treatment group did not exhibit normality. Kurtosis was computed as 3.0 with a standard error of 1.0. Therefore, the pre-treatment ATMI enjoyment scores of the treatment group were skewed negatively and were somewhat leptokurtic, exhibiting a departure from normality.

The pre-treatment ATMI enjoyment scale scores of the comparison group showed a skewness statistic of -1.2, more than twice the value of its standard error (.47). Thus, the pre-treatment ATMI enjoyment scores of the comparison group did not exhibit normality.

Kurtosis was computed as 1.3 with a standard error of .92. Therefore, the pre-treatment

ATMI enjoyment scores of the comparison group were both negatively skewed and very slightly leptokurtic, indicating a distribution that was not considered normal.

112 The pre-treatment ATMI motivation scale scores of the treatment group showed a skewness statistic of -1.1, more than twice its standard error (.52). Thus, the pre-treatment

ATMI motivation scores of the treatment group did not exhibit normality. Kurtosis was computed as .03 with a standard error of 1.0. Therefore, the pre-treatment ATMI motivation scores of the treatment group were skewed negatively and were mesokurtic, exhibiting a departure from normality.

The pre-treatment ATMI motivation scale scores of the comparison group showed a skewness statistic of -.19 with a standard error of .47. Kurtosis was computed as -.52 with a standard error of .92. Therefore, the pre-treatment ATMI motivation scores of the comparison group were both negatively skewed and nearly mesokurtic, indicating a distribution that was considered normal.

Table 9

Tests of Normality: ATMI Pre-treatment Scores by Scale and by Group

Skewness Kurtosis Scale and group Statistic Std. Er. Statistic Std. Er. Result ATMI Confidence – Treatment -2.4 .52 6.7 1.0 not normal ATMI Confidence – Comparison -.06 .47 -1.1 .92 normal ATMI Value – Treatment -1.1 .52 1.7 1.0 not normal ATMI Value – Comparison -.48 .47 -.59 .92 normal ATMI Enjoyment – Treatment -1.9 .52 3.0 1.0 not normal ATMI Enjoyment – Comparison -1.2 .47 1.3 .92 not normal ATMI Motivation – Treatment -1.1 .52 .03 1.0 not normal ATMI Motivation – Comparison -.19 .47 -.52 .92 normal

113 For all ATMI scales, pre-treatment scores in the treatment group exhibited a statistical departure from normality. Pre-treatment scores for all ATMI scales in the comparison group were normally distributed with the exception of the enjoyment scale.

Therefore, further statistical exploration of ATMI measures between the two groups was computed nonparametrically.

Pre-treatment ATMI scores, differences between groups. To examine whether initial differences existed between treatment and comparison groups on pre-treatment

ATMI inventory scores, the Mann-Whitney U statistic was computed. On pre-treatment

ATMI confidence measures, the Mann-Whitney U was computed as 104.5 (p < .01). On pre-treatment ATMI value measures, the Mann-Whitney U was computed as 107.5

(p < .01). On pre-treatment ATMI enjoyment measures, the Mann-Whitney U was computed as 76.0 (p < .001). On pre-treatment ATMI Motivation measures, the Mann-

Whitney U was computed as 93.5 (p < .01).

Therefore, based on ATMI attitude inventories administered at the outset of the study, treatment and comparison groups showed differences that were statistically significant on all four scales, specifically confidence, value, enjoyment, and motivation.

Data Obtained from the Etch-a-Sketch Digital Toy

The video game design and programming phase of the study commenced following the administration of studywide content tests and ATMI attitude inventories.

Students worked on a total of three projects, beginning with the Etch-a-Sketch. Strictly speaking, the Etch-a-Sketch was not a video game, but a digital toy.

114 Etch-a-Sketch Checkpoint Tests

At the outset of the Etch-a-Sketch project, a short checkpoint content pre-test

(Appendix D) was administered to the treatment group. The checkpoint content pre-test consisted of seven multiple choice questions relevant to mathematics content featured in the Etch-a-Sketch digital toy. Specifically, the checkpoint questions were developed based on mathematics derived from NCTM content standards, namely the numbers and operations standard, the algebra standard, or the geometry standard. The same checkpoint content test was then re-administered to the treatment group as a post-test following the completion of all Etch-a-Sketch project tasks.

Three answer possibilities existed for most questions on the checkpoint content test: correct (C), acceptable (Acc), or incorrect (I). An acceptable response was one that indicated that the student had partial understanding of how to answer the question, but missed a detail in reaching the completely correct response. Some checkpoint content questions consisted of only a correct response (C) or an incorrect (I) response.

In scoring the checkpoint content test, the shift by a student from an item response on the pre-test to an item response on the post-test was measured as follows.

1. Positive change (a shift towards a more correct response).

2. No loss (an acceptable response followed by another acceptable response, or a correct response followed by another correct response or an acceptable response).

3. No gain (an incorrect response followed by another incorrect response).

4. Negative change (a shift towards an incorrect response).

Performance from pre-test to post-test on the Etch-a-Sketch checkpoint content tests is shown in Table 10. On the numbers and operations standards, 82% of the

115 responses by treatment students exhibited either a positive change or no loss, while 19% exhibited either no gain or a negative change. On the algebra standards, 65% of the responses exhibited either a positive change or no loss; 35% exhibited either no gain or a negative change. On the geometry standards, 100% of the responses showed either a positive change or no loss. For all checkpoint content, 80% of the responses showed either a positive change or no loss, and 20% showed either no gain or negative change.

Table 10

Etch-a-Sketch Checkpoint Test Performance, Pre-test to Post-test

Change by Percentage of Responses Standards + Change No Loss No Gain - Change Numbers and operations (Q2, Q3) 21% 61% 11% 8% Algebra (Q5, Q6, Q7) 54% 11% 28% 7% Geometry (Q1, Q4) 8% 92% 0% 0% All content 32% 48% 15% 5%

Etch-a-Sketch Initial Events Tally

Following completion of the Etch-a-Sketch checkpoint mathematics pre-test, students in the treatment group commenced the Etch-a-Sketch digital toy project. As shown in Figure 6, students were provided an operational game model and given the opportunity to play individually with the Etch-a-Sketch to explore its appearance and function. Students were not able to examine any underlying programming code.

As students played with the Etch-a-Sketch model, they were asked to create an initial events list (see Appendix E) by writing down as many events as they observed.

They were informed that events described any motion, user interaction (buttons or keyboard), gamefield interaction, object interaction, scoring, or multimedia triggered

116 during play. Students were also provided an example event, “L button – press to make the tool tip move about 5 pixels “left” (west) and leave a trail,” to guide their writing.

Figure 6. Etch-a-Sketch digital toy model provided to students.

Table 11 shows the tally of initial events recorded by students during their examination of the Etch-a-Sketch digital toy model. Seventeen of nineteen students were present for the activity. The minimum number of events recorded was 4 and the maximum was 12. The mean number of events recorded for this game model was 7 (SD = 2.3).

Table 11

Tally of Events Recorded for Etch-a-Sketch Digital Toy Model

Number of Events Recorded N Min Max Mean SD Treatment group students 17 4 12 7 2.3

117 Following the creation of the initial Etch-a-Sketch events list by the treatment group, students were provided a revised Etch-a-Sketch event list (see Appendix F) by the researcher. The revised list was comprised of nine events that fully described the Etch-a-

Sketch digital toy. Students discussed the differences between their initial events list and the revised list to establish a class consensus before proceeding. The rationale for all treatment students using the same revised list was to ensure consistency in writing representations for the events during the synthesis phase of design.

Table 12

Sample Representations Provided for the Etch-a-Sketch

Representation Event Mathematical Programming

#1 L button – press to Move the tool tip 5 pixels parallel PD make the tool tip move to the x-axis in the direction of SETH 270 about 5 pixels “left” negative x with the pen down FD 5 (west) and draw a trail

#5 If the tool tip If the x coordinate of the tool tip In the backpack of the moves left and meets exceeds the x coordinate of the tool tip turtle, at the the west red barrier, left boundary, then undo the Rules tab: then the tip bounces action of the button: move the back tool tip 5 pixels “backwards” WHEN THIS (parallel to the x-axis in the XCOR < -142 direction of positive x) DO THAT BK 5

#9 When you press Lines are erased. Coordinates of CLEAN the Clear button, the the tool tip remain unchanged. screen erases the (Freeze Background drawn lines and the so that the Etch frame tool tip stays put is not erased)

118 Etch-a-Sketch Representations

Moving from the analysis phase to the synthesis phase, treatment group students were asked to write mathematical and programming code representations for each Etch-a-

Sketch event (see Appendix F). Students were provided an example representation of each type, a total of three events as shown in Table 12. Events were scored according to their relative degree of correctness: -1 for a missing or mostly incorrect representation; 0 for a partially correct representation; and +1 for a mostly correct or entirely correct representation. Combined representation scores were then computed for each student.

Etch-a-Sketch representation ratings by type. Table 13 shows descriptive statistics for representations ratings for the Etch-a-Sketch model. Of the 18 students present, mathematical representations rated a minimum of 5 and a maximum of 8 (on a range of -8 to 8 possible), with a mean of 7.2 (SD = 1.3). Programming representations rated a minimum of 1 and maximum of 9 (on a range of -9 to 9 possible), with a mean of

5.9 (SD = 2.7). Combined representation ratings ranged from a minimum of 8 to a maximum of 17 (on a range of -17 to 17 possible) with a mean of 13.1 (SD = 3.5).

Table 13

Ratings of All Representations Recorded for Etch-a-Sketch

Rating of Representations Representation N Min Max Mean SD Mathematical representations, 18 4 8 7.2 1.3 Events 1-8 (-8 to +8) Programming representations, 18 1 9 5.9 2.7 Events 1-9 (-9 to +9) Combined total representations 18 8 17 13.1 3.5 (-17 to +17)

119 Etch-a-Sketch representation ratings by standard. Events were then grouped by NCTM standards to determine ratings by type of mathematical representation recorded as shown in Table 14. Events 1-4 addressed moving the Etch-a-Sketch drawing tool tip up, down, left, and right and were categorized as geometry standards. Events 5-8 addressed using inequalities to set boundary conditions defining the edges of the drawing region and were categorized as numbers and operations as well as algebra standards (because the x- coordinate variable, the programming code xcor was employed in writing the inequality). Event 9 addressed no mathematics content as its function was to erase the

Etch-a-Sketch drawing region, a task strictly programming-oriented in nature.

Table 14

Ratings of Mathematical Representations (by Standard) Recorded for the Etch-a-Sketch.

Ratings Math representation (by Type) N Min (-4) Max (+4) Mean SD Numbers & operations and 18 0 4 3.3 1.9 algebra (Events 5-8) Geometry (Events 1-4) 18 1 4 3.8 .71

Etch-a-Sketch Initial Reflections and Design Plans

To gauge an affective facet of student progress in creating the Etch-a-Sketch digital toys, students were asked to assess their own work prior to and following the construction of their toys in MicroWorlds EX. Prior to their development of the Etch-a-

Sketch, students were asked to complete an Initial Reflections and Design Plans document

(see Appendix G). The document consisted of an explanatory sentence asking students to reflect on the events, the mathematical representations, and the programming code they

120 wrote for their game. Students were then asked to respond to open ended prompts that asked them to, (a) comment on their successes, (b) comment on their challenges, (c) sketch the appearance of their planned toy, and (d) describe modifications they planned from the digital toy model.

Statements regarding successes. In response to the successes prompt, all students who had been present during the event analysis and representations activities (N=18) made statements indicating that they experienced success on some aspect of their efforts in writing events and creating representations for the events. Comments included those addressing general feelings about the work, such as, “I am very comfortable with this. It made a lot of sense to me,” and statements more specific to student thinking, including,

“Figuring out the patterns for groups by only needing one of the codes or events.”

Statements regarding challenges. In response to the challenges prompt, 17 of 18 students identified one or more areas with which they experienced difficulty in writing events or representations. Most statements revealed that students felt challenged by creating mathematical representations, programming representations, or both. For example, one student wrote, “My challenges were in the mathematical translation. I am confused by x core and y core [sic]” and, “I didn't get the programing [sic] code for when the tip hits the barrier very well.” One student mentioned that listing the events posed difficulty. Only one individual wrote, “I did not have any challenges.”

Statements regarding planned modifications. In response to the modifications prompt, all 19 students (including the student who had been absent for event analysis and representation) provided commentary on how they would change the Etch-a-Sketch digital toy in order to personalize it. Seven students noted that they would add diagonal buttons,

121 with statements such as, “I will put more buttons on: UR, UL, DL, DR and move some buttons around.” A few statements, such as, “I would allow users to erase certain spaces as well as jumping to different positions,” demonstrated student interest in creating rather unique features in the game. While only three of the boys indicated that they wanted to change, “the color,” or alter the tool tip, all five girls noted special ways in which they intended to change the aesthetic appearance of their Etch-a-Sketches, including altering the color, the background, or the tool tip. One girl wrote, “I don't know if it's possible, but

I would like to make the thing shake when cleared,” and make the tool tip, “a piece [sic] of candy.” Another girl commented emphatically, "My tool tip will be a duck because ducks are awesome!” Three students indicated no intent to deviate from the model.

Design sketches revealed planned modifications for the Etch-a-Sketch, and featured both aesthetic changes (see Figure 7) and mathematical changes (see Figure 8).

Figure 7. Student sketch showing aesthetic modifications for Etch-a-Sketch.

122

Figure 8. Student sketch showing mathematical modifications for Etch-a-Sketch.

Etch-a-Sketch Digital Toy Programming

Following completion of their reflection and plans documents, treatment group students began constructing their Etch-a-Sketch digital toys. Game construction activities transpired in a constructivist environment with students working both independently and collaboratively. For guidance in producing their programming code in MicroWorlds EX, students referenced the game events and representations they had previously written, asking for help from the researcher and peers when needed. A typical, quick exchange among students and the researcher (RSCHR) follows:

RSCHR [in response to question] How do you jump? You could make it forward a bigger amount. RICHARD Oh – no – I know how to jump. You just go forward without putting pen down! DAYNE Oh yeah, that’s how you do it! RSCHR I like that. That was smart.

123 Occasional, directive instruction was provided when new tasks required explanation or an example from the instructor. The following dialogue is a transcript of direct instruction dialogue between the researcher and the group:

RSCHR [Smiling at class] Everybody! I need your attention on this next part. Stop what you’re doing. Spin your chairs. I need to see all your faces that way – for the next three minutes. OK. Look up at my screen. This is my drawing tool tip, right. Not nearly as beautiful as yours… I realize that. I’m going to move it to this leftmost barrier. Who knows how I can tell what the coordinates are? [Andrew raises hand.] Andrew? ANDREW Uh, you open its backpack. RSCHR Open its backpack. How do I open the backpack? ANDREW Right-click and Open Backpack. RSCHR Alright. Now, I’m at the state tab. What are the coordinates right here? JOE Negative 140 and… 8. RSCHR Which part of that is the x part? GROUP Negative 140. RSCHR So if I pick up this guy and move him more this way [horizontally]… RICHARD Oooo, you can do that? RSCHR Yeah, ‘cause he’s unfrozen right now. RICHARD Oh yeah yeah. RSCHR His x-coordinate becomes what? GROUP Negative 171. RSCHR Negative 171. It’s becoming more negative. Right? But what I care about is this boundary. Now as I move him, if I stay right at this boundary and I move him up and down… do you see that the x- coordinate is staying the same? GROUP [Quiet agreement] RSCHR What’s happening, though, to the y-coordinate. If I move up, the y- coordinate becomes more… GROUP Increases. RSCHR More positive or more negative? GROUP [Emphatically] More positive. RSCHR But if I go down, it becomes more… GROUP [Emphatically] Negative. RSCHR Negative. OK, but the x part is staying the same. It’s staying at negative 144. It’s probably the same for you. If you use my frame, it’s probably that if you stick it right at that barrier there, it’s probably negative 144.

124 The entire transcript of the class dialogue during Etch-a-Sketch construction is located in

Appendix H.

Model events included. Students worked on constructing their Etch-a-Sketch digital toys for approximately three, 45-minute class periods. Minimum construction requirements were to replicate the form and function of the model they previously examined. To replicate the model, students were required to construct graphic features, including drawing a background and adding a tool tip; and events that facilitated and controlled play. Table 15 lists the events students were required to include in their Etch-a-

Sketch digital toys and the number of students who included each event in their toy.

Table 15

Tally of Etch-a-Sketch Model Events Included in Digital Toys

Event type, Mathematics math or Number of students Event standards programming N who included event (1) Draw left Geometry both 19 19 (100%) (2) Draw right Geometry both 19 19 (100%) (3) Draw up Geometry both 19 19 (100%) (4) Draw down Geometry both 19 19 (100%) Numbers and (5) Left boundary both 19 19 (100%) operations, algebra Numbers and (6) Right boundary both 19 19 (100%) operations, algebra Numbers and (7) Top boundary both 19 19 (100%) operations, algebra Numbers and (8) Bottom boundary both 19 19 (100%) operations, algebra (9) Clear N/A programming 19 19 (100%)

125 Modifications included. Many students chose to deviate from the toy model, adding modifications they had planned for in their reflections and plans document as well as new enhancements they devised during the construction process.

Table 16 shows descriptive statistics for modifications added by students in the construction of their Etch-a-Sketch digital toys. Modifications were categorized by type, specifically mathematical or aesthetic. For example, adding a diagonal button to allow the tool tip to draw in the northwest direction was considered a mathematical modification.

Altering the appearance of the tool tip or the background design was considered an aesthetic modification. Many toys included both aesthetic and mathematical modifications.

Table 16

Etch-a-Sketch Modifications Constructed

Tally of modifications Modification type N Min Max Mean SD Math 19 0 8 4.2 2.3 Aesthetic 19 0 7 2.0 1.5 Total 19 2 15 6.2 3.1

Figure 9 shows a student-made toy that features both types of modifications. The mean number of mathematical modifications added by students was 4.2 (SD = 2.3); the mean number of aesthetic modifications added by students was 2.0 (SD = 1.5); and the mean number of total modifications added was 6.2 (SD = 3.1).

126

Figure 9. Etch-a-Sketch featuring aesthetic and mathematical modifications.

Etch-a-Sketch Final Reflections

Affective dimensions of student progress in creating the Etch-a-Sketch digital toys were measured post-construction in MicroWorlds EX. Students were asked to assess their work following the construction of their Etch-a-Sketch toys, via a Final Reflections document. (Appendix G). The document consisted of an explanatory sentence asking students to reflect on their successes and challenges in addition to sketching a figure of their completed toy and describing modifications they ultimately made.

Statements regarding successes. Post-construction, all students were able to identify successes in constructing their Etch-a-Sketch digital toys. Some remarked simply that, “It worked,” while others made statements about the areas in which they achieved success, such as, “I was able to understand the mathematics in programming.” Several

127 students identified specific tasks at which they felt successful, such as, “Changing the tool tip shape; making diagnol [sic] buttons; making cover page; making tool tip draw.”

Statements regarding challenges. Challenges listed by treatment group students on their post-construction Final Reflections documents were specific to actual difficulties encountered in building their Etch-a-Sketch digital toys and debugging until the games were fully functional. Ten out of the 19 students made remarks describing their difficulty with setting up the x and y inequalities for the tool tip. Students said that found it challenging, “putting in the input code for the bondarys [sic] so the tool tip will not go past the red barrier,” and, “I had trouble understanding the x, y, <, >, stuff.” Two students noted minor challenges such as freezing the background, and three students stated that they had no problems at all. Two students expressed general frustration with programming, stating, “I did not do well with MicroWorlds” and “Bad with computers.”

Figure 10. Etch-a-Sketch, with modifications, as constructed by treatment student.

128 Statements regarding implemented modifications. Students noted their modifications with all students listing at least one modification (mathematical or aesthetic) and most students noting two or more modifications. A representative statement from one student was, “I changed the tool tip to a penguin. When my penguin lands on a purple dot, it dances. My backround [sic] has penguins on it.” The student’s toy is shown in Figure

10. While the alteration of the tool tip and the penguin frame are aesthetic, the purple dots serve a different function – the student coded the penguin to responding to touching the purple color by “dancing,” a mathematical modification that required her writing code in the OnColor field of the penguin object (see Figure 11). The figure also shows how the student included all programming of xcor and ycor inequalities to constrain the penguin tool tip from drawing in the border as shown in the When This Do That field.

Figure 11. Programming code for the penguin object.

129 Data Obtained from the Frogger Video Game

Following completion of the Etch-a-Sketch digital toy, students commenced work on their second project, Frogger. The Frogger video game was intended to emulate the arcade game in which the player controls a frog and attempts to move it through traffic to a goal area on the opposite side of the screen while avoiding collisions with oncoming vehicles. The goal is to safely reach the opposite side of the screen.

Frogger Checkpoint Tests

At the beginning of the Frogger project, a multiple choice checkpoint content pre- test was administered to the treatment group. The Frogger checkpoint test consisted of 10 multiple-choice questions relevant to mathematics content featured in the Frogger video game. The Frogger checkpoint questions were developed based on mathematics derived from NCTM content standards, specifically the algebra standard, the geometry standard, or the measurement standard. The same Frogger checkpoint content test was then re- administered to the Treatment group as a post-test after all Frogger video game projects were completed.

For the Frogger checkpoint content tests, student responses were scored as a correct (C) response or an incorrect (I) response. In computing the results of the checkpoint content test, the shift by a student from an item response on the pre-test to an item response on the post-test was measured as positive change (I  C), no loss (C  C), no gain (I  I), or negative change (C  I). Prior to scoring the checkpoint tests, one question was omitted because no correct answer was listed among the response choices.

Performance from pre-test to post-test on the Frogger checkpoint content tests is shown in Table 17. On the algebra and geometry standards, 57% of the responses

130 exhibited either a positive change or no loss, while 43% exhibited no gain or a negative change. On the Geometry only standards, 51% of the responses exhibited either a positive change or no loss; 49% showed either no gain or a negative change. On the measurement standard, 83% of the responses exhibited either a positive change or no loss, with 18% showing either no gain or a negative change. For all content on the checkpoint test in its entirety, 64% of the responses demonstrated either a positive change or no loss, and 36% demonstrated either no gain or negative change.

Table 17

Frogger Checkpoint Test Performance (Pre-test to Post-test)

Change by Percentage of Students Standards + Change No Loss No Gain - Change Algebra & geometry 43% 14% 36% 7% (Q1, Q2, Q3, Q7) Geometry only 26% 25% 40% 9% (Q6, Q8, Q9) Measurement 23% 60% 9% 9% (Q5, Q10, Q11) All Content 32% 31% 29% 8%

Frogger Initial Events Tally

After completing the Frogger checkpoint pre-test, treatment group students commenced the Frogger video game project. Students were shown an operational video game model (see Figure 12) and invited to play individually with the Frogger game to explore its function. Underlying game code was not accessible to the students.

As students played with the Frogger video game model, they were asked to create an initial events list by writing down as many events as they observed. Students were not

131 provided an example event to guide their writing because they already possessed experience in writing events for the Etch-a-Sketch digital toy.

Figure 12. Frogger video game model.

Table 18 shows the tally of initial events recorded by students during their examination of the Frogger video game model. Eighteen of nineteen students were present for the activity. The minimum number of events recorded was 4 and the maximum was 26.

The mean number of events recorded for this game model was 11 (SD = 5.7).

Table 18

Tally of Events Recorded for Frogger Video Game Model During Game Analysis

Number of Events Recorded N Min Max Mean SD Treatment group students 18 4 26 11 5.7

132 Frogger events recorded by students included varying degrees of specificity.

Events addressing player control of the frog included statements such as, “When right arrow clicked, Frogger move right,” and “When you press  key, frog moves West 10 pxls.” Events addressing the collision between the frog and moving obstacles featured statements including, “When Frogger hits the bus he ‘splats’,” and “When frog has same quardenets [sic] as an elephant, motercycle [sic], red car or bus, frog goes to bottom and splat comes up.”

Many students applied what they had learned from constructing the Etch-a-Sketch video toy to Frogger. Some of these events fit well, for instance, setting the heading of the

“frog” and moving it incrementally. One student applied his understanding of creating top and bottom boundary conditions in Etch-a-Sketch to Frogger: he created a top boundary condition to identify the y-coordinate value defining the entry to the goal area at the top of the playfield. Other students noted that a bottom boundary condition must have existed to prevent players from “cheating” by moving down the screen and reappearing in the goal area at the top. However, a few students wrote misconceived events derived from Etch-a-

Sketch, including left and right boundary conditions. For example, one student wrote

“when frog > x280, frog bounces bk 10 pxls,” a statement constraining the frog from moving beyond the rightmost border of the playfield. Thus, not all events included in the event list composed by each student were necessary for the video game to function as shown in the game model.

Frogger Representations

Following the creation of the initial events recorded by the treatment group for the

Frogger game, students worked collaboratively to create a revised list of 18 events that

133 described the video game in its entirety. They discussed the differences between their initial events lists and established a class consensus for the revised list. This method of revising the event list differed from the method employed during the Etch-a-Sketch project. During Etch-a-Sketch, revision of the event list was directed by the researcher; during Frogger, it was directed by the students with the researcher serving as a recorder and mediator.

Some events that were finalized by the treatment group described events not actually featured in the game model, but deemed necessary by the group. For example, the model allowed the player to “cheat” by moving the frog backwards into the goal area (i.e., circumventing interaction with the obstacles). However, many students felt that players should be prevented from performing this action. They responded by writing an event that prevented the backwards cheat.

Treatment group students were then directed to write mathematical and programming code representations for each Frogger event. Two representations for events from different task classes were provided to the treatment group as shown in Table 19.

Although students were asked to write their own representations, free form discussion and idea sharing among students was permitted. Events were then scored for each student according to their relative degree of correctness: -1 for no attempt at writing a representation or a mostly incorrect representation; 0 for a partially correct representation; and +1 for a mostly correct or entirely correct representation. Mathematical, programming and combined representation scores were then computed for each student in the treatment group.

134 Table 19

Sample Representations Provided for the Frogger Video Game

Representation Event Mathematical Programming

#7 One row of traffic Elephants are moving SETH 270 (elephants) moves left parallel to x-axis in –x (W) at constant speed direction. In the elephant’s (maybe 10 pixels per ONCLICK field: second?) This traffic is Rate = distance per time FD 1 (set to FOREVER) the slowest of all lanes. = 10 pixels/sec

Some coordinate on In Frogger’s #12 Frog death: First field: Frogger’s body is the ONTOUCHING way for Frogger to die is ANNOUNCE [SPLAT!] same as some coordinate to hit an obstacle. RESET of an obstacle. PLAY

Frogger representation ratings by type. Table 20 shows descriptive statistics for representation ratings recorded by the 17 Treatment students present for the Frogger video game model. For mathematical representations, the minimum possible score was -16 and the maximum was 16. Mathematical representations written by students rated a minimum of 2 and a maximum of 16 with a mean of 11.6 (SD = 4.3). For programming representations, the minimum possible score was -15 and the maximum was 15.

Programming representations written by students rated a minimum of -2 and maximum of

15, with a mean of 10.4 (SD = 5.6). For mathematics and programming representations combined, the minimum possible score was -31 and the maximum was 31. Combined representations written by students rated a minimum of 0 and a maximum of 31 with a mean of 22.0 (SD = 9.6).

135 Table 20

Representations (by Type) Recorded for Frogger

Relative Rating of Representations Representation (by type) N Min Max Mean SD Mathematical representations, Events 1-15,18 17 2 16 11.6 4.3 (min=-16, max=16) Programming representations, Events 1-10, 12-13, 15, 16, 18 17 -2 15 10.4 5.6 (min=-15, max=15) Combined total representations 17 0 31 22.0 9.6 (min=-31, max=+31)

Frogger representation ratings by standard. Frogger events were then grouped by NCTM content standards to determine ratings by type of mathematical representation as shown in Table 21. Events 1-6, 12, and 13 were categorized as Geometry representations. These events consisted of moving the frog in the cardinal directions, setting his initial starting coordinates, returning him to his starting coordinates following a collision, and recognizing a collision between the frog and an obstacle. Events 7-11 were categorized as geometry as well as measurements. These events consisted of setting the heading and speed of obstacles in the traffic lanes and spacing them for optimum game play. Events 14, 15, and 18 were categorized as geometry as well as algebra. These events consisted of constraining playfield boundaries, winning by cheat, and winning legally.

(Events 16 and 17 did not consist of mathematical content.)

136 Table 21

Mathematical Representations (by Standard) Recorded for Frogger

Relative Rating of Representations Math Representation (by standard) N Min Max Mean SD Geometry 17 1 8 6.0 2.7 Events 1-6, 12, 13 (min=-8, max=8) Geometry and measurement 17 -1 5 3.5 1.9 Events 7-11 (min=-5, max=5) Geometry and algebra 17 0 3 2.1 1.0 Events 14, 15, 18 (min=-3, max=3)

Frogger Initial Reflections and Design Plans

Treatment group students were asked to assess their work prior to and following the construction of their video games, for the purpose of measuring an affective component of their progress. Prior to the development of their Frogger video games, students completed an Initial Plans document (Appendix G) as they had done for the Etch- a-Sketch. Students used this document to remark on successes and challenges they had experienced while writing and representing Frogger events. They also used their reflections and plans document to sketch plans for their video games and note any modifications they intended to incorporate.

Statements regarding successes. With regard to their successes, all students who were present (N=18) made a statement indicating that they had experienced success on some aspect of their efforts in writing events and crafting representations for the events.

Three students stated that understood “everything,” while others selected more narrow concepts such as feeling successful at “figuring out cheats.” Two students specifically

137 stated that they had done well with the mathematical translations, and three mentioned that were successful with the programming codes.

Statements regarding challenges. In response to the challenges prompt, 16 of 18 students identified at least one area that had challenged them. Four students mentioned concerns about programming. Another four students noted that they have various difficulties in planning how obstacles collisions would be handled in the game. The possibility of a frog-obstacle collision was a new task class for Frogger that differentiated it from the Etch-a-Sketch. Another student remarked that he was challenged by, “Solving the constant speed of all moving objects.” This also defined a new task class for Frogger, objects in constant motion. Three students wrote that they had not encountered challenges in developing the Frogger events and representations.

Figure 13. A Frogger sketch recorded in one student’s Initial Plans document.

138 Statements regarding modifications. In response to the modifications prompt, 17 of 18 students present wrote detailed comments about how they planned to adapt Frogger to their own creative visions. As students wrote, many stated aloud that the Frogger video game presented far more opportunities for them to personalize the setting, characters, and game play (compared with Etch-a-Sketch). Students wrote that they would implement modifications such as, multiple levels, portals, “many randomly moving trees,” “a platypus instead of a frog,” and “a row of spaceships.” Further, design sketches revealed that students intended to incorporate rich details in their Frogger video games (see Figure

13 and Figure 14.

Figure 14. Frogger sketch showing planned design for a “Shark Drop” game.

Students also renamed their games to reflect their planned content, with names including, “Uber Frogger,” “Candy Caper,” “The Atomic Lumberjack,” “Froggin

139 Impossible,” and “Platypus Run.” Observations corroborated their enthusiasm for the progression from the digital toy to the more flexible arcade video game project.

Frogger Video Game Programming

Following completion by the treatment group of the Initial Plans documents, students began work constructing their Frogger video games. As before, construction activities transpired in a constructivist environment with students working both independently and collaboratively. Students used documents from their design journals, specifically the revised events list and the mathematical and programming code representations, to assist them in producing their game code. The researcher and peer members of the class provided help in programming and troubleshooting coding errors as needed. Rarely was directive instruction required or provided as students found they were usually able to resolve their own problems or invoke the aid of a friend.

Game model events included. Students worked on constructing their Frogger video games for approximately seven, 45-minute class periods. Minimum construction requirements were to replicate the form and function of the video game model they previously examined. To replicate the game model, students were required to construct graphic elements, including drawing a traffic scene or other setting for the action, a frog or other character, and obstacle characters. Students could employ premade backgrounds and characters available in MicroWorlds EX, or they could create their own artwork. Students were also required to include events that facilitated and controlled game play. Table 22 lists the events students included, based on the revised event list, in their Frogger video games, along with the number of students who included each event in their game.

140 Table 22

Events from Revised List that were Included in Frogger Video Games

Event type, Number of Mathematics math or students who Event standards programming N included event (1) Move frog N Geometry both 19 18a (95%) (2) Move frog E Geometry both 19 19 (100%) (3) Move frog W Geometry both 19 19 (100%) (4) Move frog S Geometry both 19 18b (95%) (5) Set frog starting Geometry both 19 19 (100%) coordinates; start game (6) Reset after collision Geometry both 19 18 (95%) (7) Traffic row #1 Geometry, measurement both 19 19 (100%) (8) Traffic row #2 Geometry, measurement both 19 19 (100%) (9) Traffic row #3 Geometry, measurement both 19 19 (100%) (10) Traffic row #4 Geometry, measurement both 19 16 (84%) (11) Traffic spacing Geometry, measurement math 19 19 (100%) (12) Collision, frog hits Geometry both 19 19 (100%) obstacle (13) Collision, obstacle Geometry both 19 15 (79%) hits frog (14) Playfield Geometry, algebra both -- -- boundaries *excluded (15) Prevent win by Geometry, algebra both 19 5 (23%) cheat (move backwards) (16) Prevent win by N/A programming 19 19 (100%) cheat (freeze frog) (17) Prevent win by cheat of holding down N/A neither -- -- arrow keys *excluded (18) Win Geometry, algebra both 19 19 (100%)

(a) One student intentionally did not include the north direction of motion. (b) One student intentionally did not include the south direction of motion.

141 Modifications included. Table 23 shows descriptive statistics for modifications added by students in the construction of their Frogger video games. Modifications were categorized by type, specifically mathematical or aesthetic. For example, adding commands that caused obstacles to move with a circular motion or in random directions or with different speeds were mathematical modifications. Animating obstacles, writing a backstory on a splashpage, or building multiple levels were considered aesthetic modifications. The mean number of mathematical modifications added by students was

1.2 (SD = 1.2); the mean number of aesthetic modifications added by students was 8.4

(SD = 2.9); and the mean total number of modifications added was 9.6 (SD = 3.5).

Table 23

Frogger Modifications Constructed

Modifications Frogger Modifications by Type N Min Max Mean SD Mathematics 19 0 5 1.2 1.3 Aesthetic 19 5 16 8.4 3.0 Total 19 5 18 9.6 3.5

Frogger Final Reflections

Affective dimensions of student progress in creating the Frogger video games were measured post-construction. Students were asked to assess their work following the construction of their Frogger games, via a Final Reflections document. (Appendix G).

Statements regarding successes. Post-construction, all students present (N=18) were able to identify successes in constructing their Frogger video games. Many expressed general enthusiasm about their finished games with statements including, “My idea was

142 really creative,” and “My frogger was adorable!” Others were more specific about the aesthetics of their games, with one student pointing out that she did well with, “Designing very cool pages ex: cows on my win page.” Some students mentioned general successes with programming, specifically, “I really started to enjoy the concepts of microworlds,” and “I was able to work with the computer better.” Several students noted specific programming success including, “I programmed each turtle with the hit command,” “The part I thought was easy was the drive command,” “The rainbow thieves couldn’t hurt each other,” (due to use of a TOUCHEDTURTLE command) and “my game side scrolls left to right.”

Statements regarding challenges. Challenges listed by treatment group students on their post-construction Final Reflections documents identified actual difficulties encountered in building their Frogger video games and debugging until the games were fully functional. Eight of the 18 students made remarks describing difficulty with using the hit command and ensuring that the obstacles were spaced so that they did not run into each other. In this regard, one boy remarked, “I had trouble placing all the obstacles perfectly.” Similarly, one of the girls described trouble with, “random hit actions,” and another noted challenges with, “making sure the obstacles don’t collide.” Seven students described being challenged by programming, commands, or the drive command.

However, no one stated an overall sense of being challenged by designing and programming Frogger.

Statements regarding modifications. All students present (N=18) noted several modifications, both mathematical and aesthetic, in describing how their games departed from the video game model. A representative statement from a student who created the

143 Lumberjack Attack game (see Figure 15) was, “I had no lanes; I had my enemies move a random direction; I had a no-man's land (the river) with a bridge to get through; I had my enemies move a random distance under 15.” Students noted modifications to their scenes, such as changing the traffic environment to a park or an ocean. They also described changes to their main character, changing the frog to a tennis ball, or a ladybug, as well a changes to their obstacles, changing the vehicles to weather objects, gorillas, and even invisible turtles. Many students detailed their use of multiple levels, splash pages, and win pages.

Figure 15. Completed “Lumberjack Attack” video game.

Lastly, three students described their departure from the bottom-to-top crossing orientation: two used left-to-right orientations and one used a top-to-bottom orientation

(see Figure 16). Interestingly, the latter student disabled the “up” arrow in the drive

144 command to prevent the player from cheating by moving up the playfield and returning at the bottom in the goal area.

Figure 16. Completed “Shark Drop” video game.

Data Obtained from Tamagotchi Virtual Pet Video Game

Following completion of the Frogger video game, students commenced work on their third game project, Tamagotchi Virtual Pet (Tamagotchi). Tamagotchi was intended to emulate the virtual pet toys made popular in Japan, in which the player manages the

“life” of an electronic being. The goal is to keep the Tamagotchi pet alive, by feeding it when it is hungry, by hugging it when it is sad.

Tamagotchi Checkpoint Tests

At the beginning of the Tamagotchi project, a multiple choice checkpoint content pre-test was administered to the treatment group. The Tamagotchi checkpoint test

145 consisted of 7 multiple-choice questions relevant to mathematics content featured in the

Tamagotchi video game. The Tamagotchi checkpoint questions were developed based on mathematics derived from NCTM content standards, specifically the algebra standard, or the probability standard. The same Tamagotchi checkpoint content test was then re- administered to the treatment group as a post-test after all Tamagotchi video game projects were completed.

For the Tamagotchi checkpoint content tests, student responses were scored as a correct (C) response, and acceptable response (Acc) or an incorrect (I) response. In computing the results of the checkpoint content test, the shift by a student from an item response on the pre-test to an item response on the post-test was measured as positive change (I  C, I  Acc, or Acc  C), no loss (Acc  Acc, or C  C), no gain (I  I), or negative change (C  I, Acc  I, or C  Acc).

Table 24

Tamagotchi Checkpoint Test Performance (Pre-test to Post-test)

Change by Percentage of Responses Standards + Change No Loss No Gain - Change Algebra 15% 70% 8% 7% (Q1, Q2, Q3, Q4, Q6) Probability 5% 45% 40% 11% (Q5, Q7) All Content 12% 62% 17% 8%

Performance from pre-test to post-test on the Tamagotchi checkpoint content tests is shown in Table 24. On the algebra standards, 84% of the responses exhibited either a positive change or no loss, while 16% exhibited no gain or a negative change. On the

146 probability standards, 50% of the responses exhibited either a positive change or no loss;

50% showed either no gain or a negative change. For all content on the checkpoint test in its entirety, 74% of the responses demonstrated either a positive change or no loss, and

26% demonstrated either no gain or negative change.

Tamagotchi Initial Events Tally

Following completion of the Tamagotchi checkpoint pre-test, treatment group students commenced the Tamagotchi video game project. Students were shown an operational video game model (Figure 17) and invited to play individually with the

Tamagotchi game to explore its appearance and function. Students were not able to examine underlying game code.

Figure 17. Tamagotchi Virtual Pet video game model.

147 As students played with the Tamagotchi video game model, they were asked to write an events list describing the game, as they had done previously for the Etch-a-Sketch and Frogger games. Students were not provided example events because they already possessed experience in writing events from the previous two video game projects.

Table 25 shows the tally of events recorded by students during their examination of the Tamagotchi video game model. Eighteen of nineteen students were present for the activity. The minimum number of events recorded was 4 and the maximum was 26. The mean number of events recorded for this game model was 10 (SD = 5.2).

Table 25

Tally of Events Recorded for Tamagotchi Video Game Model During Game Analysis

Number of Events Recorded N Min Max Mean SD Treatment group students 19 4 26 10 5.2

Tamagotchi events recorded by students included varying degrees of recognition regarding game events and operation. Most students noticed that pressing the Live button brought the Tamagotchi pet to life, but did so with varying degrees of mathematical specificity. For example, one student wrote the event, “Live = pet begins to live (comes alive),” while another added more specificity with, “press live to reset petstats,” and another was still more specific, writing, “Live = Pet size 1 / happiness [sic] 20 / hunger

20.” Only some students included in their event lists the change in Tamagotchi pet variables with the passage of time. One student listed among his events an observation regarding only one time-controlled variable, happiness, writing, “Every 10 seconds

148 happiness – 5.” However, another student noted a more complete, though less specific, set of trends: “When you wait, his hunger increases,” and “When you wait, his happiness decreases.” Almost all students correctly commented on the function of the Tamagotchi button functions in the model video game, namely Feed and Hug. For example, one student wrote in her event list, “Hug increases happiness,” and “feed decreases hunger and increases size.” Some students played with the game enough to notice more finely nuanced details about feeding the Tamagotchi, with comments such as, “when you press feed and size is 121 the size won’t go up.” Students also noticed the Tamagotchis made announcements regarding their level of hunger or happiness. An example listed by one student was, “when hunger is 80 annonce [sic] “Boy I’m hungry!” Many students also noticed that the Tamagotchis issued announcements when fed or hugged. Towards the end of their event lists, students wrote events dealing with the conditions for Tamagotchi death: “When hunger > 100 he dies from hunger.” Only a few students remarked on the

Tamagotchi shape associated with its level of happiness, and almost none noticed that the

Tamagotchi could die of sadness. Almost all students listed a Stop event near the end of the event lists.

Tamagotchi Representations

Following the tally of events recorded by the treatment group for the Tamagotchi, students worked collaboratively to create a revised list of 11 events which described the

Tamagotchi video game in its entirety (see Appendix I). They discussed the differences between their initial events lists and established a class consensus of the revised list. This method of revising the event list of was student-directed with the researcher serving only as a recorder and mediator.

149 Students stated that they felt that the events list for the Tamagotchi pet featured events that were somewhat different from those previously encountered in Etch-a-Sketch and Frogger. Unlike the directional, motion-based games they had worked with in the first two games projects, the Tamagotchi pet featured variables whose values changed with time and whose values could be altered according to button presses by the player. Also, the Tamagotchi changed shape depending on its emotional state (i.e., its happiness level).

Because the task classes were so different for the Tamagotchi video game, the worked example representations were especially important models (see Table 26). Following examination of the example representations, treatment group students were directed to write mathematical and programming code representations for each Tamagotchi event.

Table 26

Example Representations Provided for the Tamagotchi Video Game

Representations Event Mathematical Programming

In the tama’s ONTICK field: Hunger variable is SETHUNGER HUNGER + 5 #2 Tama becomes increased by a hungrier quantity, (i.e., 5 units) Set to 50 “ticks” (tenths of one second)

In Procedures panel If the hunger variable TO FEED is greater than some IF HUNGER > 0 minimum value, [SETHUNGER HUNGER - 5] #3 Feed Tama subtract a quantity END (i.e., 5 units) from hunger Place FEED procedure in Instruction field of Feed button

150 Treatment group students were asked to write their own representations for the

Tamagotchi event; however, free form dialogue and sharing between students was permitted. Events were scored according to their relative degree of correctness: -1 for no attempt at writing a representation or a mostly incorrect representation; 0 for a partially correct representation; and +1 for a mostly correct or entirely correct representation.

Combined representation scores were then computed for each treatment student.

Table 27

Representations Recorded for Tamagotchi Virtual Pet

Relative Rating of Representations Representation N Min Max Mean SD Mathematical representations, Events 1, 2, 2a, 2b, 3, 3a, 4, 4a, 5, 5a 16 -3 10 4.8 4.5 (min=-10, max=10) Programming representations, Events 1, 2, 2a, 2b, 3, 3a, 4, 4a, 5, 5a, 6 16 -5 11 3.9 4.8 (min=-11, max=11) Combined total representations 16 -8 21 8.8 8.3 (min=-21, max=+21)

Tamagotchi representation ratings by type. Table 27 shows descriptive statistics for representation ratings recorded by the 16 Treatment students present for the

Tamagotchi video game model. For mathematical representations, the minimum possible score was -10 and the maximum was 10. Mathematical representations written by students rated a minimum of -3 and a maximum of 10 with a mean of 4.8 (SD = 4.5). For programming representations, the minimum possible score was -11 and the maximum was

11. Programming representations written by students rated a minimum of -5 and

151 maximum of 11, with a mean of 3.9 (SD = 4.8). For mathematics and programming representations combined, the minimum possible score was -21 and the maximum was 21.

Combined representations written by students rated a minimum of -8 and a maximum of

21 with a mean of 8.8 (SD = 8.3).

Tamagotchi representation ratings by standard. Tamagotchi events were then grouped by NCTM content standards to determine ratings by type of mathematical representation as shown in Table 28. Events 1, 2a, 2b, 3, 4a, and 5 were categorized as algebra representations. These events consisted of setting variable values for the

Tamagotchi pet (e.g., setting the initial values), modifying a variable value (e.g., hugging the pet, thereby increasing happiness) and making an announcement in response to variable values (e.g., announcing “I need some attention,” when happiness equals zero).

Table 28

Mathematical Representations (by Type) Recorded for Tamagotchi Virtual Pet

Relative Rating of Representations Math Representation (by Type) N Min Max Mean SD Algebra Events 1, 2a, 2b, 3, 4a, and 5 16 -2 6 3.4 2.3 (min=-6, max=6) Algebra and measurement 16 -1 2 1.1 1.0 Events 2 and 4 (min=-2, max=2) Algebra and probability 16 -2 2 0.3 1.6 Events 3a and 5a (min=-2, max=2)

Events 2 and 4 were categorized as algebra and measurement. These events caused the Tamagotchi pet to get hungrier and to become sadder at a specified time interval.

Events 3a and 5a were categorized as algebra and probability. These events consisted of

152 making announcements with a predetermined probability when the Tamagotchi was fed or hugged. (Event 6, the Stop event, did not consist of mathematical content.)

Tamagotchi Initial Reflections and Design Plans

To measure an affective component of student progress, the treatment group was asked to assess their work prior to and following the construction of their video games.

Before developing their Tamagotchi video games, students completed a reflections and plans document (see Appendix G) as they had done previously for the Etch-a-Sketch digital toy and the Frogger video game. Students used this document to remark on successes and challenges they had experienced while writing and representing Tamagotchi events. They also used their reflections and plans document to sketch plans for their video games and note any modifications they intended to incorporate.

General statements were made by two students who said that they were successful,

“designing the game,” and one student who felt successful at “everything.” Thirteen students specifically stated that they were successful in making Tamagotchis, characters, or the background. One student reflected that he felt had experienced, “growth in technology,” and a different student mentioned he was successful with the mathematical translations.

Statements regarding challenges. In response to the challenges prompt, 16 of 18 students identified at least one area that had challenged them. Eight students stated general concerns about writing programming code. The new task class of using variables to

153 indicate Tamagotchi attributes was challenging for some students. Five students said that they were challenged by variables, writing statements such as, “Making the variables go up and down over time is hard,” and “the felings [sic] are challenging for me.” One student mentioned difficulty with the “mathematic translation.” Only two students wrote that they had not experienced challenges in developing the Tamagotchi events and representations.

Statements regarding modifications. In response to the modifications prompt, all

18 students present wrote extensive details about how they planned to alter the

Tamagotchi video game model to personalize the virtual pet. As with Frogger, students were vocal in brainstorming and expressing their thoughts about how their Tamagotchi creations would look and function.

All students indicated their intent to draw and name their own Tamagotchi and situate it in a different scene, such as creating exotic bird in a rainforest or, “a sheep that lives in a meadow.” Two students wrote that they still wanted to maintain a many aspects of the game model with statements such as, “I’m using hunger and happiness, but I also have a heaven button.” Others described plans to maintain similar variable constructs (one variable that increases and a second variable that decreases over time), but rename the variables and redesign the pet and scene. This was the case with one student who wrote that his planned modifications were, “intelligence goes up; wealth goes down; space background; alien pet.” However, most students described or drew detailed plans (see

Figure 18) that were significantly different from the game model, with many featuring multiple variables and creative animations. As one girl wrote, “I don’t have a petsize; I

154 made a new variable, dirtiness; I have a bunny instead of a ballon [sic]; I have different pics come up when I do things like feed it.”

Figure 18. Design plans for Tinki pet, including variables and aesthetic elements.

Other students developed multiple scenes for their Tamagotchis so that the pets could “live” in different settings. For example, one girl drew three Tamagotchi characters and two scenes and wrote the variables she planned to employ with her multi-pet video game, namely, “play; take a bath; swim; sleep; boredom; stinkiness; activeness; health.”

(See Figure 19.)

Half the boys (N=7) in the treatment group wrote plans that included battle, training, or weapons, with themes ranging from World War II to “Pixelmon” (i.e.,

Pokemon) training. One boy wrote that his modifications consisted of, “Hunger that increases over time strength from winning battles; Platinum for money; focus that

155 decreases over time; energy decreases from battles; size from getting strength.” One boy mentioned an idea that no one else did, specifically, “I think I will have buttons connecting to Frogger and Etcha-Sketch [sic].”

Figure 19. Planned Tamagotchi with multiple pets, scenes, and interacting variables.

156 Tamagotchi Video Game Programming

Following completion by the treatment group of the reflections and plans documents, students began work constructing their Tamagotchi video games. Continuing with the format that had been implemented during the two previous games, video game construction activities transpired in a constructivist environment with students working both independently and collaboratively. Students used their events and representations documents to assist them in producing their game code in MicroWorlds EX.

However, due to the increased complexity of constructing the Tamagotchi pet and the new variable task classes with which students had no prior experience, students found that the representations they had written were not sufficiently complete nor supportive in programming their Tamagotchi pets. To assist students in successfully bridging between writing representations on paper and programming their games on the computer, the researcher employed a code completion instructional strategy (van Merriënboer &

Krammer, 1987). The researcher provided the programming code for the Tamagotchi game model, called Balloonichi, with some complete procedures and some gaps (e.g., missing code) where other procedures were needed. The class worked together, with researcher guidance, to use the code provided as a model for completing the missing code.

(See Appendix I for the Balloonichi game model code completion worksheet.) The code completion strategy was successful in helping students understand how to program their

Tamagotchi video games.

Treatment group students then resumed programming their Tamagotchi pets, with the researcher and peer members providing assistance in programming and

157 troubleshooting coding errors as needed. One-on-one troubleshooting was required more frequently than on previous games, due to the highly customized undertaken by students.

Game model events included. Students worked on constructing their Tamagotchi video games for approximately twelve, 45-minute class periods. Minimum construction requirements were to replicate the form and function of the video game model they previously examined. To replicate the game model, students were required to construct graphic elements, including drawing a setting for the action and drawing a Tamagotchi character. Students could employ premade backgrounds and characters available in

MicroWorlds EX, or they could create their own artwork using the built-in drawing tools.

Graphic elements for the Tamagotchi Virtual Pet also included indicators and interaction buttons. Indicators were text boxes that reported the changing values of the variables they employed in their video games (e.g., hunger, happiness). Interaction buttons were buttons that the player could press (e.g., feed, hug) to interact with the Tamagotchi pet and change the variable values. While the Tamagotchi game model presented two indicators and four interaction buttons, students were free to modify the model and use any number of indicators and interaction buttons as they required to construct their design plans.

Finally, students were required to include events that facilitated and controlled game play. Table 29 lists the events students included, based on the revised event list, in their Tamagotchi video games, along with the number of students who included each event in their game.

158 Table 29

Events from Revised List that were Included in Tamagotchi Video Games

Event type, Mathematics math or Number of students Event standards programming N who included event (1) Live Algebra both 19 19 (100%) (2) Increase hunger Algebra, both 19 18 (95%) or other variable over time measurement (2a) Announce hunger or Algebra both 19 12 (63%) other rising variable (2b) Dies from hunger or Algebra both 19 11 (58%) other variable (3) Feed (or other task) to Algebra both 19 19 (100%) decrease variable (3a) Announce feeding or Algebra, both 19 10a (53%) similar action probability (4) Decrease happiness or Algebra, both 19 17 (89%) other variable over time measurement (4a) Announce decrease in happiness or other falling Algebra both 19 9b (47%) variable (5) Hug (or other task) to Algebra both 19 19 (100%) increase variable (5a) Announce hug or Algebra, both 19 9c (47%) similar action probability (6) Stop N/A programming 19 17 (89%)

(a) Seven students animated feeding (or similar action); three issued an announcement. (b) One student animated the decrease in variable; eight issued an announcement. (c) Five students animated hugging (or similar action); four issued an announcement.

Modifications included. Table 30 shows descriptive statistics for modifications added by students in the construction of their Tamagotchi video games. Modifications were categorized by type, specifically mathematical or aesthetic. Mathematical modifications included: adding commands that created new variables and set their initial values; manipulating variable values; causing pets to grow or shrink in size; executing compound conditionals; using waits and distances in animation; or issuing announcements

159 when equality or inequality conditions were met. Aesthetic modifications included: animating shape changes; writing a backstory on a splashpage; drawing original character or scenic graphics; or adding interaction buttons. The mean number of mathematical modifications added by students was 15 (SD = 12); the mean number of aesthetic modifications added by students was 20 (SD = 18); and the mean total number of modifications added was 35 (SD = 27).

Table 30

Tamagotchi Modifications Constructed

Modifications Tamagotchi Modifications by Type N Min Max Mean SD Mathematics 19 0 38 15 12 Aesthetic 19 1 60 20 18 Total 19 1 80 35 27

Tamagotchi Final Reflections

Qualitative, affective dimensions of student progress in creating the Tamagotchi video games were measured post-construction. Students were asked to assess their work following the construction of their Tamagotchi games, via a Final Reflections document.

(See Appendix G).

Statements regarding successes. Post-construction, all students present (N=18) were able to identify successes in constructing their Tamagotchi games. One of the young men stated that he succeeded with, “Just about everything,” while another wrote, “My creativity excelled.” One young woman said simply, “I had fun. I liked being able to create something.” Two students wrote that they felt successful with programming in

160 general, and several others identified specific coding successes including, “making buttons, making background, changing screens, and most codes.” Some were excited to share the details of their variables in action, writing statements including, “I successfully got to ‘lull’ my guy and ‘power’ him up;” and “It was easy for me to set the rating and to make the ontick work.” Many students also described their work in the aesthetic realm, noting successes such as, “The design of the feelings of the Tamagotchi,” and “creating different characters and finishing the animation.” Researcher observations corroborated students’ self-reported successes as well as their general enthusiasm for creating the final video game project.

Statements regarding challenges. Challenges listed by treatment group students

(N=18) on their post-construction Final Reflections documents related mostly to programming. Fifteen of eighteen students present remarked generally on programming challenges – “I had trouble with the commands” – or specifically on programming challenges – “I had trouble setting a procedure to change shape.” Only one student mentioned “not much” in the way of challenges. In general, students wrote fewer challenges than they had documented for the Etch-a-Sketch and Frogger projects.

Observations corroborated that students were becoming more adept at programming, more capable of troubleshooting and resolving coding errors, and more capable of adapting their construction activities to content and skills they already possessed.

Statements regarding modifications. All students present (N=18) were prolific in noting their many modifications, both mathematical and aesthetic, of the Tamagotchi

Virtual Pet video game. Some noted that they had maintained the general structure of the

Tamagotchi game model, but modified the character and the names of the two variables.

161 One example was the “Alien Tamagotchi Pet” which featured an alien pet in space with intelligence that increased and wealth that decreased over time (see Figure 20).

Figure 20. Alien Tamagotchi Virtual Pet.

A similar, completed Tamagotchi game that also featured two variables, including wealth, was set in a snowy mountaintop setting. The treatment student modified the game model by creating two “turtle” objects – one turtle became a mountain goat Tamagotchi pet and one turtle became a money pile. Writing code for buttons that affected the goat and money pile simultaneously, the student reported the state of the pet’s hunger and wealth numerically through the variable indicators. Additionally, using setsize commands in the programming code, he caused physical changes to occur during the life of his pet. For example, the goat grew and the money pile shrank whenever a player pressed the “Feed” button, because food costs money and induces growth.

162 Another student created a World War II version of the Tamagotchi, noting his inventive modifications: “I made a splash page. I made different shapes and variables. My background was different.” As seen in Figure 21, he created a Fear variable that increased over time, but could be decreased by pressing the Rally Troops button. The student also created two variables, Ammunition and Soldiers, that decreased over time, but could be increased by pressing the Reinforcements or Bring in extra ammunition buttons, respectively. The student was also creative in crafting his Stop button which he renamed

Cease-fire.

Figure 21. World War II Battle (Tamagotchi Virtual Pet).

As shown in Figure 22, one young woman also employed three variables that changed over time, but she created many additional buttons the player could press to affect variable values. For her Bunny Tamagotchi, she stated, “I took out petsize and added

163 Dirtiness. On my feed command I had it show a picture of my bunny eating a carrot. I added dance using the repeat command, also a wash and a mud bath. I also gave it an eat candy command.” This student also included several announcements that were issued when variables equaled set values.

Figure 22. Bunny Tamagotchi.

Figure 23, shows the BanMan in the Jungle Tamagotchi, which the student designed to feature six variables that changed over time and could be changed as a result of pressing interaction buttons. The student wrote, “I had banananess, energy, age, and awesomeness rating [as well as petsize and hunger]. I had 11 Tamagotchi shapes. They showed how many bananas he had.” For his Get a Banana button, the student created a special procedure employing conditionals: if BanMan possessed zero bananas, pressing

Get a Banana changed his shape so that he held one banana; if BanMan possessed one

164 banana already, pressing Get a Banana changed his shape so that he held two bananas.

The student’s Eat button reversed the shape changes in similar fashion, one banana at a time. Additionally, the student applied inequality content he learned in Etch-a-Sketch to constrain the motion of BanMan when the Dance button was pressed: he wrote a procedure so that BanMan set a new heading whenever a certain xcor or ycor value was exceeded, forcing him to dance only within a central area of the playfield.

Figure 23. BanMan in the Jungle (Tagmagotchi Virtual Pet).

One student who created, a sheep named Fluffo who lived in a meadow, implemented 80 modifications in her Tamagotchi video game. Figure 24 shows one scene, the Bath scene, from a game that also included a meadow scene for playing, a rainbow scene for snacking, and a gym scene for exercising. Multiple variables, original graphics, animations, buttons and announcements offered a complex video game for players to

165 engage in. The student wrote procedures that caused buttons to affect more than one variable and trigger animations. For instance, Feed Beans caused Fluffo to lose health, lose happiness, and turn green in the face. Additionally, the student applied the OnColor command she learned in Frogger to cause Fluffo to bounce when she touched the surface of a black trampoline in the gym scene.

Figure 24. Fluffo Sheep Tamagotchi Pet.

Finally, one student who stated simply that she modified, ”Everything,” created a

Tamagotchi pet that employed complex animations in which she set shapes, coordinates, headings, forward motions, and wait times to create a dynamic player experience.

Executing the design plan she previously drew in Figure 18, the student created a Tinki

Tamagotchi that also used drop down boxes and associated procedures to allow for the selection of fruits to eat and activities for play (see Figure 25).

166

Figure 25. Completed Tinki Tamagotchi Pet.

Studywide Post-test Performance

Following completion of the entire study, all students engaged in a studywide mathematics content post-test – the same test administered as a pre-test at the outset of the study. The studywide content tests consisted of 20 questions relevant to the three video game projects undertaken during the study. Results of the studywide content post-test are shown in Table 31. Treatment group students scored a mean of 12.7 (SD = 2.3) and comparison group students scored a mean of 11.0 (SD = 1.9)

167 Table 31

Studywide Content Post-test Scores by Group

Group N Min Max Mean SD Treatment 19 9 18 12.7 2.3 Comparison 24 5 15 11.0 1.9 All participants 43 5 18 11.8 2.3

* Range of test was 0 (min) to 20 (max).

Studywide Questions Derived from Etch-a-Sketch

A subset of the studywide content questions was derived from content featured in the Etch-a-Sketch toy. Because all study participants took the studywide content tests, data were obtained for both treatment group and comparison group participants.

For pre-test to post-test performance, data on the studywide content test was collected using the same techniques employed for the checkpoint content tests. The shift by a student from an item response on the pre-test to an item response on the post-test was measured as either a positive change; no loss; no gain; or negative change.

Treatment group performance. Performance from pre-test to post-test on the studywide mathematics content addressing Etch-a-Sketch subset questions is shown for the treatment group in Table 32. On the numbers and operations standards, 100% of the treatment group responses exhibited either a positive change or no loss. On the geometry standards, 81% of the treatment group responses exhibited either a positive change or no loss while 18% showed either no gain or a negative change. On the algebra standards,

44% of the treatment group responses exhibited either a positive change or no loss, while

56% showed either no gain or a negative change. For all Etch-a-Sketch content on the

168 studywide content test, 70% of the treatment group responses demonstrated either a positive change or no loss, and 29% demonstrated either no gain or negative change.

Table 32

Studywide Content, Etch-a-Sketch Subset (Pre-test to Post-test) – Treatment Group

Change by percentage of treatment responses Etch-a-Sketch standards + Change No Loss No Gain - Change Numbers and operations (Q2, Q3) 21% 79% 0% 0% Geometry (Q1, Q4) 18% 63% 5% 13% Algebra (Q5, Q6, Q7) 28% 16% 51% 5% All content 23% 47% 23% 6%

Comparison group performance. Performance from pre-test to post-test on the studywide content addressing Etch-a-Sketch subset questions is shown for the comparison group in Table 33. On the numbers and operations standards, 96% of the comparison group responses exhibited either a positive change or no loss. On the geometry standards,

80% of the comparison group responses exhibited either a positive change or no loss while

18% showed either no gain or a negative change. On the algebra standards, 43% of the comparison group responses exhibited either a positive change or no loss, while 57% showed either no gain or a negative change. For all Etch-a-Sketch content on the studywide test in its entirety, 69% of the comparison group responses demonstrated either a positive change or no loss, and 32% demonstrated either no gain or negative change.

169 Table 33

Studywide Content, Etch-a-Sketch Subset (Pre-test to Post-test) – Comparison Group

Change by percentage of comparison responses Etch-a-Sketch standards + Change No loss No gain - Change Numbers and operations (Q2, Q3) 6% 90% 4% 0% Geometry (Q1, Q4) 15% 65% 4% 17% Algebra (Q5, Q6, Q7) 35% 8% 43% 14% All content 21% 48% 21% 11%

Treatment vs. comparison group performance. Table 34 shows treatment versus comparison group performance on the studywide mathematics content test with regard to

Etch-a-Sketch subset questions. On the numbers and operations standards, the treatment group showed positive change or no loss on 100% of the questions, while the comparison group performed similarly on 96% of the questions. On the geometry standards, the treatment group showed positive change or no loss on 81% of the questions, while the comparison group performed similarly on 80% of the questions. On the algebra standards, the treatment group showed positive change or no loss on 44% of the questions, while the comparison group performed similarly on 43% of the questions. On all content standards, the treatment group showed positive change or no loss on 70% of the questions, while the comparison group performed similarly on 69% of the questions.

170 Table 34

Studywide Content, Etch-a-Sketch Subset (Pre-test to Post-test) – All Participants

Change by percentage of responses + Change or No loss Etch-a-Sketch standards Treatment Comparison Numbers and operations (Q2, Q3) 100% 96% Geometry (Q1, Q4) 81% 80% Algebra (Q5, Q6, Q7) 44% 43% All content 70% 69%

Studywide Questions Derived from Frogger

A subset of the studywide content questions was derived from content featured in the Frogger game. Because all study participants took the studywide content tests, data was obtained for both treatment group and comparison group participants. For pre-test to post-test performance, the shift by a student from an item response on the pre-test to an item response on the post-test was measured as either a positive change; no loss; no gain; or negative change.

Treatment group performance. Performance from pre-test to post-test on the studywide content addressing Frogger subset questions is shown for the treatment group in

Table 35. On the algebra and geometry standards, 23% of the questions answered by the treatment group either a positive change or no loss, while 77% showed either no gain or a negative change. On the geometry standards, 57% of the responses exhibited either a positive change or no loss while 42% showed either no gain or a negative change. On the measurement standards, 85% of the treatment group questions exhibited either a positive change or no loss, while 16% showed either no gain or a negative change. For all Frogger

171 content on the studywide test in its entirety, 54% of the treatment group responses demonstrated either a positive change or no loss, and 46% demonstrated either no gain or negative change.

Table 35

Studywide Content, Frogger Subset (Pre-test to Post-test) – Treatment Group

Change by percentage of treatment responses Frogger standards + Change No Loss No Gain - Change Algebra and geometry (Q5, Q6, Q10) 12% 11% 63% 14% Geometry (Q12, Q13) 39% 18% 39% 3% Measurement (Q11, Q14, Q15) 11% 74% 5% 11% All content 18% 36% 36% 10%

Comparison group performance. Performance from pre-test to post-test on the studywide content addressing Frogger subset questions is shown for the comparison group in Table 36. On the algebra and geometry standards, 24% of the questions answered by the comparison group students exhibited either a positive change or no loss; 76% showed no gain or negative change. On the geometry standards, 35% of the comparison group responses exhibited either a positive change or no loss while 65% showed either no gain or a negative change. On the measurements standards, 80% of the comparison group responses exhibited either a positive change or no loss, while 20% showed either no gain or a negative change. For all Frogger content on the checkpoint test in its entirety, 48% of the comparison group responses demonstrated either a positive change or no loss, and

52% demonstrated either no gain or negative change.

172 Table 36

Studywide Content, Frogger Subset (Pre-test to Post-test) – Comparison Group

Change by percentage of comparison responses Frogger standards + Change No Loss No Gain - Change Algebra and geometry (Q5, Q6, Q10) 17% 7% 68% 8% Geometry (Q12, Q13) 25% 10% 63% 2% Measurement (Q11, Q14, Q15) 26% 54% 7% 13% All content 22% 26% 44% 8%

Treatment vs. comparison group performance. Table 37 shows treatment versus comparison group performance on the studywide content test with regard to Etch-a-Sketch subset questions. Only percentages associated with positive change or no loss are shown.

Table 37

Studywide Content, Frogger Subset (Pre-test to Post-test) – All Participants

Change by percentage of responses + Change or No loss Frogger standards Treatment Comparison Algebra and geometry (Q5, Q6, Q10) 23% 24% Geometry (Q12, Q13) 57% 35% Measurement (Q11, Q14, Q15) 85% 80% All content 49% 48%

On the algebra and geometry standards, the treatment group showed positive change or no loss on 23% of the questions, while the comparison group performed similarly on 24% of the questions. On the geometry standards, the treatment group showed

173 positive change or no loss on 57% of the questions, while the comparison group performed similarly on 35% of the questions. On the measurement standards, the treatment group showed positive change or no loss on 85% of the questions, while the comparison group performed similarly on 80% of the questions. On all content standards, the treatment group showed positive change or no loss on 49% of the questions, while the comparison group performed similarly on 48% of the questions.

Studywide Questions Derived from Tamagotchi

A subset of the studywide content questions was derived from content featured in the Tamagotchi game. Because all study participants took the studywide content tests, data was obtained for both treatment group and comparison group participants.

For pre-test to post-test performance, the shift by a student from an item response on the pre-test to an item response on the post-test was measured as either a positive change; no loss; no gain; or negative change.

Treatment group performance. Performance from pre-test to post-test on the studywide content addressing Tamagotchi subset questions is shown for the treatment group in Table 38. On the algebra standards, 79% of the treatment group responses exhibited either a positive change or no loss, while 21% of the responses showed either no gain or a negative change. On the probability standards, 61% of the treatment group responses exhibited either a positive change or no loss while 40% showed either no gain or a negative change. For all Tamagotchi content on the studywide test, 71% of the treatment group responses demonstrated either a positive change or no loss, and 28% demonstrated either no gain or negative change.

174 Table 38

Studywide Content, Tamagotchi Subset (Pre-test to Post-test) – Treatment Group

Change by percentage of treatment responses Tamagotchi standards + Change No Loss No Gain - Change Algebra (Q16, Q17, Q18) 33% 46% 12% 9% Probability (Q19, Q20) 16% 45% 32% 8% All content 26% 45% 20% 8%

Comparison group performance. Performance from pre-test to post-test on the studywide content addressing Tamagotchi subset questions is shown for the comparison group in Table 39. On the algebra standards, 61% of the comparison group responses exhibited either a positive change or no loss, while 39% of the responses showed either no grain or a negative change. On the probability standards, 43% of the comparison group students exhibited either a positive change or no loss, while 56% showed either no gain or a negative change. For all Tamagotchi content on the studywide test in its entirety, 55% of the comparison group students demonstrated either a positive change or no loss, and 46% demonstrated either no gain or negative change.

Table 39

Studywide Content, Tamagotchi Subset (Pre-test to Post-test) – Comparison

Change by percentage of comparison students Tamagotchi standards + Change No Loss No Gain - Change Algebra (Q16, Q17, Q18) 21% 40% 26% 13% Probability (Q19, Q20) 10% 33% 50% 6% All content 17% 38% 36% 10%

175 Treatment vs. comparison group performance. Table 40 shows treatment versus comparison group performance on the studywide content test with regard to

Tamagotchi subset questions. On the algebra standards, the treatment group showed positive change or no loss on 79% of the questions, while the comparison group performed similarly on 61% of the questions. On the geometry standards, the treatment group showed positive change or no loss on 61% of the questions, while the comparison group performed similarly on 43% of the questions. On all content standards, the treatment group showed positive change or no loss on 71% of the questions, while the comparison group performed similarly on 55% of the questions.

Table 40

Studywide Content, Tamagotchi Subset (Pre-test to Post-test) – All participants

Change by percentage of participating students + Change or No loss Tamagotchi standards Treatment Comparison Algebra (Q16, Q17, Q18) 79% 61% Probability (Q19, Q20) 61% 43% All content 71% 55%

Studywide Content Tests, Pre-test-to-Post-test Changes

All study participants took the studywide content tests so that data was obtained for both treatment group and comparison group participants. For pre-test to post-test performance, the shift by a student from an item response on the pre-test to an item response on the post-test was measured as either a positive change; no loss; no gain; or negative change.

176 Treatment group performance. Performance by the treatment group from pre- test to post-test on the studywide content test is shown in Table 41. Question items were grouped according to NCTM content standard and programming project.

Table 41

Studywide Content Test (Pre-test to Post-test) – Treatment Group

Change by percentage of treatment responses Standards + Change No Loss No Gain - Change Numbers and operations (Etch) 21% 79% 0% 0% Algebra (Etch, Tamagotchi) 31% 31% 32% 7% Algebra and geometry (Frogger) 12% 11% 63% 14% Geometry (Etch, Frogger) 29% 41% 22% 8% Measurement (Frogger) 11% 74% 5% 11% Probability (Tamagotchi) 16% 45% 32% 8% All standards 22% 42% 27% 8%

On the numbers and operations standards, 100% of the treatment group responses showed either a positive change or no loss. On the algebra standards, 62% of the treatment group responses exhibited either a positive change or no loss while 39% showed either no gain or a negative change. On the algebra and geometry standards, 23% of the treatment group responses exhibited either a positive change or no loss, while 77% showed either no gain or a negative change. On the geometry standards, 70% of the treatment group responses exhibited either a positive change or no loss, while 30% showed either no gain or a negative change. On the measurement standards, 85% of the treatment group responses exhibited either a positive change or no loss, while 16% showed either no gain or a negative change. On the probability standards, 61% of the treatment group responses

177 exhibited either a positive change or no loss, while 40% showed either no gain or a negative change. For all content on the studywide test in its entirety, 64% of the treatment group responses demonstrated either a positive change or no loss, and 35% demonstrated either no gain or negative change.

Comparison group performance. Performance by the comparison group from pre-test to post-test on the studywide content test is shown in Table 42. Question items were grouped according to NCTM content standard and programming project.

Table 42

Studywide Content Test (Pre-test to Post-test) – Comparison Group

Change by percentage of comparison students Standards + Change No Loss No Gain - Change Numbers and operations (Etch) 6% 90% 4% 0% Algebra (Etch, Tamagotchi) 28% 24% 35% 13% Algebra and geometry (Frogger) 17% 7% 68% 8% Geometry (Etch, Frogger) 20% 38% 33% 9% Measurement (Frogger) 26% 54% 7% 13% Probability (Tamagotchi) 10% 33% 50% 6% All standards 20% 36% 34% 10%

On the numbers and operations standards, 96% of the treatment group responses showed either a positive change or no loss, while 4% showed either no grain or negative change. On the algebra standards, 52% of the treatment group responses exhibited either a positive change or no loss while 48% showed either no gain or a negative change. On the algebra and geometry standards, 24% of the treatment group responses exhibited either a positive change or no loss, while 76% showed either no gain or a negative change. On the

178 geometry standards, 58% of the treatment group responses exhibited either a positive change or no loss, while 42% showed either no gain or a negative change. On the measurement standards, 80% of the treatment group responses exhibited either a positive change or no loss, while 20% showed either no gain or a negative change. On the probability standards, 43% of the treatment group responses exhibited either a positive change or no loss, while 56% showed either no gain or a negative change. For all content on the studywide test in its entirety, 64% of the treatment group responses demonstrated either a positive change or no loss, and 35% demonstrated either no gain or negative change.

Treatment vs. comparison group performance. Table 43 shows treatment versus comparison group performance on the studywide content test in its entirety. On the numbers and operations standards, the treatment group showed positive change or no loss on 100% of the questions, while the comparison group performed similarly on 94% of the questions. On the algebra standards, the treatment group showed positive change or no loss on 62% of the questions, while the comparison group performed similarly on 52% of the questions. On the algebra and geometry standards, the treatment group showed positive change or no loss on 23% of the questions, while the comparison group performed similarly on 24% of the questions. On the geometry standards, the treatment group showed positive change or no loss on 70% of the questions, while the comparison group performed similarly on 58% of the questions. On the measurement standards, the treatment group showed positive change or no loss on 85% of the questions, while the comparison group performed similarly on 80% of the questions. On the measurement standards, the treatment group showed positive change or no loss on 61% of the questions,

179 while the comparison group performed similarly on 43% of the questions. On all content standards, the treatment group showed positive change or no loss on 64% of the questions, while the comparison group performed similarly on 56% of the questions.

Table 43

Studywide Content Test (Pre-test to Post-test) – All Participants

Change by percentage of participating students + Change or No loss Standards Treatment Comparison Numbers and operations (Etch) 100% 96% Algebra (Etch, Tamagotchi) 62% 52% Algebra and geometry (Frogger) 23% 24% Geometry (Etch, Frogger) 70% 58% Measurement (Frogger) 85% 80% Probability (Tamagotchi) 61% 43% All standards 64% 56%

An additional method of viewing changes in pre-test to post-test performance study participants is to examine the raw scores of both treatment and performance students at the outset and the end of the study period. As shown in Table 44, the pre-test mean of the treatment group was 10.2 (out of 20), and the post-test mean of the treatment group was 12.7. At the outset of the study, some variables for study participants were found to be normally distributed, while others were not. For consistency, all statistical tests examining study participants were conducted nonparametrically. The Wilcoxon Signed Ranks test, a nonparametric t-test used for two paired groups, was used to compare the pre-test-to-post- test scores for each of the two groups. For the treatment group, the Wilcoxon Z was found to be -3.1 (p < .01), a statistically significant improvement. For the comparison group, the

180 Wilcoxon Z was found to be -3.6 (p < .001), a statistically significant improvement. For all participants, the Wilcoxon Z was found to be -4.7 (p < .001), a statistically significant improvement. The Mann-Whitney U was computed for the between groups pre-post mean shift and was found to be not statistically significant.

Table 44

Within Groups Studywide Content (Pre-test to Post-test) Scores

Pre-test Post-test Pre-Post Pre-to-Post Comparison Group N Mean* Mean* Mean ∆ Wilcoxon Z Significance Treatment 19 10.2 12.7 2.5 -3.1 p < .01 Comparison 24 9.0 11.0 2.0 -3.6 p < .001 All participants 43 9.5 11.8 2.3 -4.7 p < .001

* Range of test was 0 (min) to 20 (max).

Post-treatment ATMI Inventory Outcomes

At the completion of the study, all study participants took the ATMI inventory, the same inventory they completed at the outset of the study. Table 45 shows the post- treatment ATMI inventory scores by scale for the treatment and comparison groups. For the post-treatment ATMI confidence scale (max = 75), the treatment group scored a mean of 65.8 (SD = 5.3) while the comparison group scored a mean of 59.2 (SD = 8.5). For the post-treatment ATMI value scale (max = 50), the treatment group scored a mean of 46.3

(SD = 4.6) and the comparison group scored a mean of 43.6 (SD = 4.9). For the post- treatment ATMI enjoyment scale (max = 50), the treatment group scored a mean of 43.4

(SD = 8.0) and the comparison group scored a mean of 38.2 (SD = 6.9). For the post- treatment ATMI motivation scale (max = 25), the treatment group scored a mean of 21.5

181 (SD = 3.7) and the comparison group scored a mean of 18.6 (SD = 4.3). While ATMI post-treatment scores were high for all study participants, scores for the treatment group were higher than scores for the comparison group on all four scales.

Table 45

ATMI Post-treatment Scores by Scale and by Group

ATMI Post-treatment Scores Scale and group N Min Max Mean SD ATMI Confidence – Treatment 19 49 / 75 70 / 75 65.8 / 75 5.3 ATMI Confidence – Comparison 24 38 / 75 70 / 75 59.2 / 75 8.5 ATMI Value – Treatment 19 33 / 50 50 / 50 46.3/ 50 4.6 ATMI Value – Comparison 24 32 / 50 50 / 50 43.6/ 50 4.9 ATMI Enjoyment – Treatment 19 28 / 50 50 / 50 43.4 / 50 8.0 ATMI Enjoyment – Comparison 24 22 / 50 49 / 50 38.2 / 50 6.9 ATMI Motivation – Treatment 19 13 / 25 25 / 25 21.5 / 25 3.7 ATMI Motivation – Comparison 24 10 / 25 25 / 25 18.6 / 25 4.3

Studywide Pre-treatment-to-Post-treatment ATMI Scores (Within Groups)

Tables 46, 47, 48, and 49 show the pre-treatment to post-treatment ATMI inventory scores by scale and by group. In general, scores for all scales and groups remained stable during the study, decreasing slightly from pre-treatment to post-treatment.

One exception was that the confidence scale score by comparison group increased slightly over the course of the study.

Because some pre-treatment ATMI inventory scores exhibited normal distributions and others did not, nonparametric tests were performed for all ATMI inventory measures for consistency and ease of comparison. For within group comparisons on all four ATMI

182 scales – confidence, value, enjoyment, motivation – the Wilcoxon Z statistic was computed. No statistical differences were found between the pre-treatment mean scores and the post-treatment mean scores within each group.

Table 46

Within Group Studywide ATMI Confidence Pre-treatment to Post-treatment Scores

ATMI Confidence Scores Pre- Post- Wilcoxon treatment treatment Pre-Post Pre-to-Post Group N Mean* Mean* Mean ∆ Z score Treatment 19 66.0 65.8 -0.2 -.08 (p = .94) Comparison 24 58.5 59.2 0.7 -.50 (p = .62) All participants 43 61.8 62.1 0.3 -.32 (p = .75)

* Range of inventory was 0 (min) to 75 (max).

Table 47

Within Group Studywide ATMI Value Pre-treatment to Post-treatment Scores

ATMI Value Scores Pre- Post- Wilcoxon treatment treatment Pre-Post Pre-to-Post Group N Mean* Mean* Mean ∆ Z score Treatment 19 47.3 46.3 -1.0 -1.5 (p = .14) Comparison 24 43.6 43.2 -0.4 -.57 (p = .57) All participants 43 45.2 44.6 -0.6 -.1.3 (p = .19)

* Range of inventory was 0 (min) to 50 (max).

183 Table 48

Within Group Studywide ATMI Enjoyment Pre-treatment to Post-treatment Scores

ATMI Enjoyment Scores Pre- Post- Wilcoxon treatment treatment Pre-Post Pre-to-Post Group N Mean* Mean* Mean ∆ Z score Treatment 19 45.8 43.4 -2.4 -1.5 (p = .13) Comparison 24 38.6 38.2 -0.4 -.43 (p = .67) All participants 43 41.8 40.5 -1.3 -1.4 (p = .18)

* Range of inventory was 0 (min) to 50 (max).

Table 49

Within Group Studywide ATMI Motivation Pre-treatment to Post-treatment Scores

ATMI Motivation Scores Pre- Post- Wilcoxon treatment treatment Pre-Post Pre-to-Post Group N Mean* Mean* Mean ∆ Z score Treatment 19 22.9 21.5 -1.4 -1.8 (p = .08) Comparison 24 18.8 18.6 -0.2 -.02 (p = .98) All participants 43 20.6 19.9 -0.7 -1.1 (p = .29)

* Range of inventory was 0 (min) to 25 (max).

Studywide Pre-treatment-to-Post-treatment ATMI Scores (Between Groups)

For between group comparisons on all four ATMI scales – confidence, value, enjoyment, motivation – the Mann-Whitney U was computed on the pre-post mean differences. No statistically significant differences were found between treatment and comparison groups with regard to their change in scores on any of the four scales.

184 Other Correlational Relationships

Statistical computations were made to search for other possible relationships between variables in this study. Only one such relationship was found, specifically, a gender-related statistical difference that appeared among the young women in the treatment group. On the ATMI enjoyment scale, girls exhibited a decrease in scores between pre-treatment and post-treatment that was statistically significant (Mann-Whitney

U = 2.0, p < .05). However, since there were so few girls (N=5) in the treatment group, further research with a larger study group should be conducted to obtain better data relevant to this relationship.

185 CHAPTER 5

DISCUSSION

This study evaluated the mathematics content learned and attitudes exhibited by students engaged in the design and construction of video games over several months. It emerged from recent research demonstrating the development of complex reasoning and problem-solving skills (Fadjo, Chang, Hong, & Black, 2010) and mathematics concepts

(Kafai, 1995) by youth engaged in video game design. During the study, treatment group students analyzed video games for their mathematical events, synthesized the mathematics of video game events, and programmed functional games.

Summary of Research Questions

Three research questions guided this study. The questions addressed learning mathematics content, transferring mathematics content knowledge, and mathematics attitude as follows.

Question 1, Mathematics of Video Game Design and Construction

Question 2, Lateral Transfer of Mathematics Content Knowledge

186 choice mathematics content test, how does the performance of middle school students who are engaged in video game design and construction compare with the performance of students of similar math abilities who are not engaged in video game design and construction?

Question 3, Attitudes Toward Mathematics

To investigate these questions, 6th and 7th grade middle school students engaged in designing and constructing three video games using the MicroWorlds EX environment.

Treatment students designed and constructed (a) an Etch-a-Sketch (b) a Frogger and (c) a

Tamagotchi Virtual Pet. They also took mathematics content tests and attitude inventories pre-treatment and post-treatment. Nineteen treatment group students and 24 comparison group students took part in the study.

Question 1a – Mathematics of Video Game Design and Construction: Analysis

During game analysis, treatment group students successfully identified most events for all three video games as measured by tallies of their initial events. They demonstrated proficiency in identifying approximately the correct number of mathematical events that defined each game.

187 Analyzing Video Game Events

For each video game project, treatment group students were asked to analyze the video game model and generate an initial events list – all mathematical events they believed defined the complete game. For each treatment student, a tally of all events on the initial events list was documented.

Following their initial events list, the researcher and class collaborated to generate a revised events list before proceeding to synthesizing representations. The revised events list was aggregated and streamlined from individual student events lists, and represented group consensus of the mathematical events that completely defined the game model.

Treatment students were able to discern most events from the three video game models. The mean number of events tallied on their initial lists was very close to the number of events agreed upon for the revised events list.

Tallies of Initial Events by Project

In the Etch-a-Sketch digital toy project, students recorded on their initial events list a mean number of 7 events (SD = 2.3). The revised events list generated 9 events. For the Frogger video game project, students recorded on their initial events list a mean number of 11 events (SD = 5.7). The revised events list generated 18 events, two of which were not required to make the game. Many students grouped an entire task class together

(e.g., writing “press arrow keys to move the frog”) instead of specifying subtasks (e.g., writing “press left arrow key to move frog west”) in their events lists. This underrepresented the number of initial events recorded by students. Finally, in the

Tamagotchi video game project, students recorded on their initial events lists a mean

188 number of 10 events (SD = 5.2). The revised events list generated 11 events to completely produce the game.

Question 1b – Mathematics of Video Game Design and Construction: Synthesis

During game synthesis for each video game project, treatment group students examined their revised events list and wrote representations for each event. Students were provided example representations for each game and then asked to write both a mathematical and a programming representation for each event. Mathematical representations were further categorized by mathematical standard.

Both mathematical and programming representations were then rated with a -1, 0 or 1 based on their degree of accuracy. Treatment group students achieved positive mean scores in or near the upper quartile of the score range on all mathematical and programming representations written for the three game projects. However, representations scores did not rise with increasing experience, but instead decreased slightly with increasing levels of game complexity and the introduction of new task classes.

Synthesizing Representations in Etch-a-Sketch

For the Etch-a-Sketch, students demonstrated success in writing both types of representations, but were more adept in writing mathematical than programming representations. This may have been due to their inexperience in writing programming code at the early stages of the study. They produced combined representation scores with a mean of 13.1 (SD = 3.5) on a possible range of -17 to 17. Their programming representation scores had a mean of 5.9 (SD = 2.7) on a possible range of -9 to 9.

189 Mathematical representation scores had a mean of 7.2 (SD = 1.3) on a possible range of

-8 to 8.

Deconstructing the Etch-a-Sketch mathematical representations further, students were very successful in writing mathematical representations for all NCTM standards relevant to the toy. Upon reflection, most treatment students stated that they felt challenged by writing representations, mathematical or programming or both. The eight mathematical events in the game addressed content of the numbers and operations and algebra standards as well as the geometry standards. Mathematical representations scores addressing numbers and operations and algebra had a mean of 3.3 (SD = 1.9) on a possible range of -4 to 4. Mathematical representations scores addressing geometry had a mean of

3.8 (SD = .71) on a possible range of -4 to 4.

Synthesizing Representations in Frogger

For Frogger, students in the treatment group demonstrated some degree of success in writing both types of representations, and were equally adept in writing mathematical and programming representations. They produced combined representation scores with a mean of 10.4 (SD = 5.6) on a possible range of -15 to 15. Upon student reflection, treatment students stated that their increasing experience made them feel successful writing representations, but that they were especially challenged by writing programming code addressing the new task classes in Frogger. Their programming representation scores had a mean of 5.9 (SD = 2.7) on a possible range of -9 to 9. Their unfamiliarity with writing programming code addressing collisions (OnTouching), reaching the goal zone

(OnColor), and setting relative speeds of objects (FD and WAIT) reduced their

190 programming representation scores. Mathematical representation scores had a mean of

11.6 (SD = 4.3) on a possible range of -16 to 16.

Deconstructing Frogger mathematical representations further, students were successful in writing mathematical representations for all standards relevant to the Frogger game. The 16 mathematical events in the game addressed content of the geometry, measurement, and algebra standards. Mathematical representations scores addressing geometry-only standards had a mean of 6.0 (SD = 2.7) on a possible range of -8 to 8.

Mathematical representations scores addressing geometry paired with measurement had a mean of 3.5 (SD = 1.9) on a possible range of -5 to 5. Mathematical representations scores addressing geometry paired with algebra had a mean of 2.1 (SD = 1.0) on a possible range of -3 to 3.

Synthesizing Representations in Tamagotchi Virtual Pet

For the Tamagotchi Virtual Pet, representation scores were in the positive range and showed that students demonstrated some degree of success in writing both types of representations. However, scores were closer to the zero point than on previous projects, implying that students were generally less successful in writing representations for

Tamagotchi pet than for Etch-a-Sketch and Frogger. Early in the Tamagotchi project, inexperience in working with variables may have affected student ability to write both mathematical and programming representations for the many variable-related events in

Tamagotchi. Upon student reflection, most treatment students stated that they felt challenged by writing representations for both mathematics and programming code for the new task classes addressing variables. However, many students also expressed enthusiasm about pursuing the new concepts encompassed in the Tamagotchi pet because of the

191 tremendous creative opportunities they presented. Students in the treatment group produced combined representations scores with a mean of 8.8 (SD = 8.3) on a possible range of -21 to 21. It is worth noting that the range of total representations scores for the

Tamagotchi project was -8 to 21, indicating a large variation among students in successfully writing representations. At least one student scored well below the zero point while another wrote every representation correctly. Programming representations scores for the Tamagotchi had a mean of 3.9 (SD = 4.8) on a possible range of -11 to 11.

Mathematical representations scores had a mean of 4.8 (SD = 4.5) on a possible range of

-10 to 10.

Deconstructing Tamagotchi representations further, students were somewhat successful in writing mathematical representations for all standards relevant to the

Tamagotchi game. The 10 mathematical events in the game addressed content of the algebra, measurement, and probability standards. Mathematical representations scores addressing algebra-only standards had a mean of 3.4 (SD = 2.3) on a possible range of -6 to 6. Mathematical representations scores addressing algebra paired with measurement had a mean of 1.1 (SD = 1.0) on a possible range of -2 to 2. Mathematical representations scores addressing algebra paired with probability had a mean of 0.3 (SD = 1.6) on a possible range of -2 to 2.

Question 1c – Mathematics of Video Game Design and Construction: Programming

Game construction was the process of transforming the revised events list into functional programming code resulting in a playable video game. Most treatment group students successfully included all video game model events from the revised events lists in

192 the construction of their video games. Statements by students who did not include certain video game model events indicated their intentions to deviate from the model for specific reasons, as opposed to inability in programming the events.

Treatment group students also had the option of modifying their video games, altering or adding to the revised list of events when constructing their games. As they progressed through programming the three game projects, students added increasingly larger numbers of modifications, both mathematical and aesthetic, as they personalized their video games.

Programming Etch-a-Sketch

The first of three projects that treatment students programmed was the Etch-a-

Sketch. The Etch-a-Sketch was a computer-based analogue of the physical drawing toy.

Etch-a-Sketch model events. Constructing the Etch-a-Sketch digital toy entailed nine events agreed upon by treatment group students. These nine events addressed the following.

1. Geometry standards, consisting of four events for drawing in the four cardinal directions.

2. Numbers and operations standards along with algebra standards, consisting of four boundary condition events for constraining the tool tip from drawing in the frame.

3. One programming-only event for cleaning off the drawing (i.e., the Etch-a-

Sketch “shake” feature).

All 19 students (100%) included each of the nine events in their Etch-a-Sketch toys. This suggested success in learning and applying the geometry, numbers and operations, and algebra concepts from which the Etch-a-Sketch was constructed. Upon

193 reflection on their Etch-a-Sketch construction activities, more than half the treatment group (10 out of 19 students) described challenges with programming inequalities to constrain the tool tip.

Etch-a-Sketch modifications. For the Etch-a-Sketch digital toy, in the reflections and plans design documents, treatment students wrote of their intent to add mathematical modifications, such as creating buttons for allowing diagonal drawing; and aesthetic modifications, such as changing the color of the frame or shape of the tool tip. The mean number of mathematical modifications created for the Etch-a-Sketch was 4.2 (SD = 2.3), and the mean number of aesthetic modifications created was 2.0 (SD = 1.5). The mean number of total modifications created for the Etch-a-Sketch was 6.2 (SD = 3.1). One student noted difficulty in thinking of new ways to modify the Etch-a-Sketch. However, considering that the digital toy model consisted of only nine events, it was surprising that treatment students were able to include, on average, six modifications for their toys.

Programming Frogger

The second of the three projects that treatment students programmed was Frogger.

Frogger was a computer-based analogue of the popular arcade game that challenged players to safely hop a frog through traffic.

Frogger model events. Constructing the Frogger video game entailed 18 events agreed upon by treatment group students. During construction, students excluded two events, deeming them observations more than programmable events. The 16 resulting events addressed the following.

1. Geometry-only standards, consisting of eight events. Four events enabled hopping the frog in the four cardinal directions; two events were used for setting the frog’s

194 starting coordinates; and two events were used to identify coordinate convergence during a collision between a frog and an obstacle. Nearly 100% of the students included all of the geometry-only events. Students who did not include all of the geometry-only events either used an alternative programming command to handle collisions or intentionally eliminated a directional command to prevent the player from moving in certain directions.

2. Geometry and measurement standards consisting of five events for moving the rows of traffic and keeping traffic obstacles sufficiently spaced apart. Again, nearly 100% of the students included all geometry and measurement events, although some eliminated the fourth row of traffic for aesthetic reasons.

3. Geometry and algebra standards consisting of two events, one for reaching the goal zone to win the game and another for preventing a win via backward cheat (i.e., moving the frog backwards into the goal zone without advancing in the forwards direction). All students included the win event. Only 23% included the prevention of the backward cheat, an event that required programming an inequality similar to code learned in Etch-a-Sketch. However, the video game model did not feature this event and some students said they thought it was more interesting for players if the cheat were allowed.

4. One programming-only event, preventing a cheat by picking up and moving an unfrozen frog to the goal zone. This event was included by 100% of the students.

Upon reflection on their Frogger construction activities, members of the treatment group identified challenges with handling the new task class of collisions. But the general consensus of students was that the construction process was becoming fun, and that they felt enthusiastic about their emerging skills and the freedom to create their own variations of the Frogger video game.

195 Frogger modifications. For the Frogger video game, in the reflections and plans design documents, treatment students wrote of their intent to add mathematical modifications, such as altering the speed or heading of obstacles; and aesthetic modifications, such as changing the frog and obstacle shapes, or the game orientation. The mean number of mathematical modifications created for the Frogger was 1.2 (SD = 1.3), and the mean number of aesthetic modifications created was 8.4 (SD = 3.0). The mean number of total modifications created for the Frogger was 9.6 (SD = 3.5). Thus, most modifications made to Frogger were aesthetic, not mathematical, in nature. However, the aesthetic modifications caused the finished video game projects to take on vastly different appearances and game play experiences.

Programming Tamagotchi Virtual Pet

The third of the three projects that treatment students programmed was the

Tamagotchi Virtual Pet. The Tamagotchi was a computer-based analogue of the hand-held toy requiring players to tend to the needs of a virtual pet.

Tamagotchi model events. Constructing the Tamagotchi video game entailed 11 events agreed upon by treatment group students. The 11 events addressed the following.

1. Algebra-only standards consisted of six events involving setting variables, and increasing and decreasing their values. Events included bringing the Tamagotchi to life, feeding (or performing an alternate task) to decrease a variable, and hugging (or performing an alternate task) to increase a variable. One hundred percent of the students included these algebra-only events in their Tamagotchi pets. Three other events addressing algebra-only standards consisted of announcing hunger (or other increasing variable), dying from hunger (or other variable) and announcing happiness level (or other decreasing

196 variable). For each of these events, only half of the treatment students included the event in their Tamagotchi pets. Students stated various reasons for not including these model events in their games, with most reasons being aesthetic in nature. Many students said they found it annoying for the Tamagotchi to be constantly making announcements; some also said that they simply didn’t like the idea of the Tamagotchi “dying.”

2. Algebra and measurement standards consisted of two events, one for increasing hunger (or other variable) over time and one for decreasing happiness (or other variable) over time. Within the treatment group, 95% of the students included the increasing variable event and 89% of the students included the decreasing variable event.

3. Algebra and probability standards consisted of two events, announcing feeding

(or similar) and announcing hugging (or similar) with a frequency determined by a random number. For each of these events, only half of the treatment students included the event in their Tamagotchi pets. Students had difficulty in working with the programming code for generating random numbers, which was associated with making announcements.

During discussions with the treatment group, students indicated that their departure from the Tamagotchi video game model reflected their intent to develop their games in their own unique ways, not a failure to understand how to replicate the video game model.

Their extensive modifications of the Tamagotchi pet seemed to support these stated intentions.

Tamagotchi modifications. For the Tamagotchi game, in the reflections and plans design documents, treatment students wrote of their intent to add mathematical modifications, such as altering variables and variable interactions; and aesthetic modifications, such as changing the Tamagotchi shape or animating changes in

197 Tamagotchi appearance. The mean number of mathematical modifications created for the

Tamagotchi was 15 (SD = 12), and the mean number of aesthetic modifications created was 20 (SD = 18). The mean number of total modifications created for the Tamagotchi was 35 (SD = 27). Most students made extensive modifications to their Tamagotchis, both mathematically and aesthetically. Additionally, many students wrote procedures that caused multiple variables to adjust simultaneously and activate physical changes in the appearance or behavior of the pet.

Support of the Hypothesis Regarding Video Game Design and Construction

At the outset of the study, it was hypothesized that middle school students who engaged in video game design would learn age-appropriate math concepts (e.g., algebra, geometry, and measurement) as prescribed by standards outlined by the National Council of Teachers of Mathematics (NCTM, 2000). The successful performance of treatment group students in (a) identifying mathematical events during video game analysis; (b) writing mathematical and programming representations for events during video game synthesis; and (c) programming video games featuring both game model events and extensive modifications (including mathematical) during video game construction, supported this hypothesis and mirrored outcomes from previous, related studies (Kafai,

1995; McCoy, 1996; Papert, 1996).

Question 2 – Lateral Transfer of Mathematics Content Knowledge

Mathematics content tests were administered during the study to measure the degree to which lateral transfer of mathematics concepts occurred from the computer-

198 programming context to the traditional paper-and-pencil context. Two types of mathematics content tests were administered, studywide tests and checkpoint tests.

Three checkpoint tests were administered to treatment group students. Tests were specific to content of each of the three video game projects and were administered prior to video game analysis and following video game construction for each video game project.

Each checkpoint test consisted of seven to ten multiple-choice items addressing standards- based, mathematics content relevant to the three video game projects.

The studywide mathematics test was administered at the outset and close of the entire study to all participants. The studywide test was a 20-item, standards-based, multiple-choice mathematics test with questions derived from content featured in the three video game projects.

Performance of mathematics content tests was evaluated by two methods. The checkpoint tests were evaluated via score shifts, examining the shift toward a more correct or more incorrect response on each item. The studywide test was evaluated via raw scores, with a theoretical minimum of 0 and a theoretical maximum of 20. Additionally, the subset of studywide test items associated with each video game project was extracted and the score shifts of these items were also examined.

Question 2a – Lateral Transfer of Mathematics, Within Treatment Group

Lateral transfer of mathematics content knowledge was evaluated for the treatment group. It was considered with regard to content standards and to each individual video game project.

Lateral transfer by content standard (treatment group). With regard to mathematics content standards, treatment students demonstrated better than average score

199 shifts (≥ 50%) on all checkpoint tests and most studywide test items. Students showed very strong performance (82% checkpoint / 100% studywide) on numbers and operations standards. They also exhibited strong performance (84% checkpoint / 79% studywide) on algebra standards addressing variable manipulation; and moderate performance (65% checkpoint / 44% studywide) on algebra standards addressing inequalities and boundary conditions. On geometry standards addressing coordinates and directions, they performed extremely well (100% checkpoint / 81% studywide). On geometry standards addressing the area of a circle as well as parallel and perpendicular lines, they performed moderately well (51% checkpoint / 57% studywide). Students struggled (57% checkpoint / 23% studywide) on combined algebra and geometry standards addressing equations of lines and the Pythagorean Theorem. On measurement standards addressing rate, students performed strongly (83% checkpoint / 85% studywide). Finally, on probability standards, students performed moderately well (50% checkpoint / 61% studywide).

Lateral transfer by project (treatment group). With regard to individual video game projects, treatment students demonstrated better than average score shifts (≥ 50%) on checkpoint tests in their entirety and on project-related subsets of the studywide tests.

On the Etch-a-Sketch related tests, treatment students demonstrated either a positive change or no loss on 80% of the responses on the checkpoint test and 70% of the responses on the Etch-a-Sketch content of the studywide test. They demonstrated either a positive change or no loss on 64% of the responses on the Frogger checkpoint test and

54% of the responses on the Frogger content of the studywide test. Finally, treatment students demonstrated either a positive change or no loss on 74% of the responses on the

200 Tamagotchi checkpoint test and 71% of the responses on the Tamagotchi content of the studywide test.

Question 2b – Lateral Transfer of Mathematics, Between Groups Comparison

At the outset of the study, it was asked how the performance of middle school students who engaged in video game construction (treatment group) would compare with the performance of students of similar math abilities who were not engaged in video game construction (comparison group). Two types of comparisons were made to answer this question. The first type of comparison was via mean raw scores of each group on the studywide pre-test and post-test. Both treatment and comparison groups made statistically significant raw score gains on pre-to-post administrations of the studywide test. However, the difference between the score gains of the groups was not statistically significant. The second type of comparison was via pre-test-to-post-test score shifts on the studywide tests by group. On the studywide mathematics content tests, the treatment group exhibited positive gain or no loss on 64% of the responses while the comparison group exhibited positive gain or no loss on 56% of the responses. In terms of the performance shift between pre-test and post-test responses on the studywide content test, treatment group students exhibited greater shifts on all content areas except one. Specifically, the treatment group exhibited greater shifts than the comparison group on numbers and operations, algebra-only, geometry-only, measurement, and probability standards. The treatment group exhibited a nearly equal shift as the comparison group on combined algebra and geometry standards.

Studywide shifts by raw score. Treatment group students (N=19) scored a mean of 10.2 on the studywide pre-test and a mean of 12.7 on the post-test, producing a mean

201 shift of 2.5. The Wilcoxon within group change Z value of -3.1 was statistically significant

(p < .01) for treatment students. Comparison group students (N=24) scored a mean of 9.0 on the studywide pre-test and a mean of 11.0 on the post-test, producing a mean shift of

2.0. The Wilcoxon within group change Z value of -3.6 was also statistically significant (p

< .001) for comparison students. The Mann-Whitney U was computed for the between groups pre-post mean shift and found not to be statistically significant.

Studywide shifts by mathematics content standards. On the numbers and operations standard, treatment students showed a positive gain or no loss on 100% of the responses, while the comparison group performed similarly on 96% of the responses. On the algebra standard, treatment students exhibited positive gain or no loss on 62% of the responses, while the comparison group performed similarly on 52% of the responses. On the geometry standards, the shift was 70% (treatment) versus 58% (comparison). On the measurement standards, the shift was 85% (treatment) versus 80% (comparison). And on the probability standards, the shift was 61% (treatment) versus 43% (comparison). Only on the combined algebra and geometry standards did comparison group students slightly outperform treatment group students, with the treatment students achieving positive gain or no loss on 23% of the responses, and the comparison students performing similarly on

24% of the responses.

Support of the Hypothesis Regarding Transfer of Math Content Knowledge

At the outset of the study, it was hypothesized that treatment students who engaged in video game design and construction would improve performance on tests of standards- based mathematics content. The positive shift (≥ 50%) by treatment group students on all

202 pre-project to post-project checkpoint tests in (a) performance on each mathematics content standard, and (b) overall performance, supported this hypothesis. Further, the positive shift in performance (≥ 50%) was replicated on most standards as measured on the pre-treatment and post-treatment studywide tests. There was variation in how well performance improvements were retained over time. When comparing the score shifts on the checkpoint tests with analogous content on the studywide tests, it appeared that the most-recently learned content was retained best.

It was also hypothesized that students engaging in video game design and construction would achieve higher score gains on a studywide test of standards-based mathematics content than peers of similar mathematics abilities who did not engage in video game design and construction as compared on pre- and post-treatment administrations of the test. Both the treatment and comparison groups were shown to have statistically significant positive score shifts, with the treatment group producing a larger shift (∆ = 2.5) than the comparison group (∆ = 2.0). However, the larger shift of the treatment group scores was not found to be statistically significant, and thus the hypothesis was not supported. With regard to content standards, the treatment group outperformed the comparison group on numbers and operations, algebra, geometry, measurement, and probability. While the percentage differences between the groups was not high, it did appear that there was some additional lateral transfer from programming to the traditional context for these standards. However, there was little difference between the groups in performance on the combined algebra and geometry content standard. Both groups exhibited poor performance, and the content conveyed through MicroWorlds EX programming did not transfer well to the traditional context for this subject matter. As

203 noted by Littlefield et al. (1988), successful transfer of mathematics content from the programming to the traditional context may require explicit bridging by the teacher.

Question 3 – Attitudes Toward Mathematics

At the outset of the study, it was asked how the attitude of middle school students toward mathematics could be characterized prior to and after constructing video games, and how these attitudes would compare with students of similar math abilities who did not construct games. To answer this question, all study participants completed a 39-item attitude inventory, the Attitude Towards Mathematics Inventory (ATMI) (Tapia & Marsh,

2004). The ATMI is a valid and reliable, 5-point, Likert-style instrument for measuring mathematics attitude on four scales, confidence (14 items), value (10 items), enjoyment

(10 items), and motivation (5 items). Because some score distributions exhibited normality and other did not on the pre-treatment administration of the ATMI, all statistics comparing

ATMI scores were computed nonparametrically.

Approximately 7 months elapsed between ATMI administrations. During this time, treatment group students engaged in the design and construction of three video games, while comparison group students engaged in no other study-related activities.

Question 3a – Attitudes Within the Treatment Group

On all four scales of the ATMI, treatment group students exhibited extremely favorable attitudes toward mathematics. These attitudes held relatively stable during the course of the study, decreasing only slightly from pre-treatment to post-treatment. The

Wilcoxon Z score indicated no significance in the decreases.

204 Examining treatment group (N=19) attitudes for each scale, the ATMI confidence scale (max = 75) score showed a mean of 66.0 pre-treatment and 65.8 post-treatment. On the ATMI value scale (max = 50) the score had a mean of 47.3 pre-treatment and 46.3 post-treatment. On the ATMI enjoyment scale (max = 50), treatment group students scored a mean of 45.8 pre-treatment and 43.4 post-treatment. The ATMI motivation scale

(max = 25) score showed a mean of 22.9 pre-treatment and 21.5 post-treatment.

Additionally, the positive attitudes measured by the ATMI were corroborated by

(a) favorable comments made by students toward video game design and construction during the treatment period, (b) student success in creating each video game, and (c) the time and effort invested in developing extensive modifications for the games.

Question 3b – Attitudes Between Groups

On all four scales of the ATMI, comparison group students exhibited favorable attitudes toward mathematics, although these attitude scores were slightly lower than those of the treatment group. Comparison group attitudes held relatively stable during the course of the study, increasing slightly on the confidence scale and decreasing slightly on all other scales from pre-treatment to post-treatment. The Wilcoxon Z score indicated no significance in the attitude changes of the comparison group. Further, the Mann-Whitney

U showed no significant difference in the pre-to-post changes between groups.

On the ATMI confidence scale (max = 70), treatment group students (N=19) decreased 0.2 pre-treatment to post-treatment, while comparison group students (N=24) increased 0.7. On the ATMI value scale (max = 50), treatment group students decreased

1.0 pre-treatment to post-treatment, while comparison group students decreased 0.4. On the ATMI enjoyment scale (max = 50), treatment group students decreased 2.4 pre-

205 treatment to post-treatment, while comparison group students decreased 0.4. On the ATMI motivation scale (max = 25), treatment group students decreased 1.4 pre-treatment to post- treatment, while comparison group students decreased 0.2.

Rejection of the Hypothesis Regarding Attitudes Toward Mathematics

At the outset of the study, it was hypothesized that, within the treatment group, attitudes toward mathematics scores would increase from pre- to post-treatment measurements. For the treatment group, attitudes scores on all four ATMI scales were high, both pre-treatment and post-treatment, decreasing slightly between administrations of the inventory. Wilcoxon Z scores showed the decrease had no statistical significance.

Thus, the hypothesis positing that treatment group disposition scores would rise during the study was rejected.

Lastly, it was hypothesized that post-treatment attitudes toward mathematics would be higher for treatment group students than for comparison group students. It was found that treatment group attitude scores were higher than comparison group scores for both pre-treatment and post-treatment administrations of the ATMI. However, no statistically significant differences were found between groups with regard to their changes in scores on any of the four scales. While the hypothesis was supported, it could not be concluded that the higher attitude scores of the treatment group were associated with the treatment intervention.

Conclusions

Based on the quantitative and qualitative data collected and analyzed in this study, several conclusions can be reached. Conclusions must be qualified with regard to the

206 study participants engaged in this study. Because study participants were of high mathematics ability and high socioeconomic status, conclusions about their experiences and their performance may not be generalizable to broader audiences. Further, consideration must be given to the fact that students in the treatment group chose to participate in the math enrichment course knowing that they would spend several months engaged in video game design and construction as part of this study.

1. Middle grade students similar to those in this study can successfully analyze

(identify) the events defining game play in a video game or toy. These students can recognize mathematics, appropriate to their grade level, that is inherent in the design of simple video games and toys. Events include motion, collisions, and scoring.

2. Provided worked examples, middle grade students similar to those in this study can successfully synthesize (represent) video game and toy events in both mathematical and programming forms. Representations include writing and coding (a) boundary conditions using inequalities, (b) coordinate locations and identification of coordinate convergence, (c) directional headings, (d) uniform linear motion, (e) variable changes, and

(f) probability-based consequences.

3. Provided a video game or toy model, middle grades students similar to those in this study can successfully computer program a similarly functioning game or toy. In programming their games and toys, they can demonstrate mastery of mathematical content knowledge acquired through analyzing and synthesizing game or toy events. They can also customize the game or toy by adding their own aesthetic and mathematical modifications. Further, they can apply knowledge acquired from previous video game and toy programming projects in inventive ways during the construction of new projects.

207 4. Middle grades students, similar to those in this study who engaged in the design and construction of video games and toys, can transfer mathematical knowledge to the traditional, multiple-choice test format. Because students engaged in this study took both a traditional math course and a math enrichment course (addressing video game and digital toy production), it is not possible to attribute the transfer exclusively to the content learned in the math enrichment course.

Additionally, without explicit bridging by the teacher between instructional contexts, transfer may not be extensive nor permanent. Additionally, the instruction of mathematics via video game and toy design and construction may not differ in efficacy than other instructional methods.

5. Middle grades students similar to those in this study who exhibit favorable attitudes towards mathematics maintain those attitudes, with no statistically significant differences, following several months working in the design and construction of video games. Additionally, students engaged in video game and toy design and construction can accurately identify successes and challenges when reflecting on their work.

6. Video games and toys are inherently built from multiple mathematics concepts.

Students engaged in the design and construction of video games and toys must employ these multiple concepts to create their games. Curriculum designers and teachers who plan to incorporate video game and toy instruction in mathematics courses must plan carefully to ensure correlation between game and toy projects and math concepts that are secure or emerging for learners.

In summary, this study demonstrated that curriculum addressing design and construction of video games and toys can be viable, cognitively and affectively, for

208 instructing age-appropriate, standards-based mathematics content. However, learning may not be easily transferred to other contexts such as traditional, paper-and-pencil, multiple- choice tests. Finally, high achievement in mathematics and high socioeconomic status of the participants may cause these conclusions not to be generalizable to other populations.

Limitations

This study possessed several limitations that must be taken into account when evaluating its applicability or extensibility to other populations and settings. These limitations follow.

1. Limited sample sizes. The small number of students in the treatment group

(N=19) and in the comparison group (N=24) make it challenging to draw statistically sound conclusions that can be generalized.

2. High mathematics achievement of participants. Because study participants possessed an average achievement test score in mathematics of 92.5%, they were not representative of typical student populations. Their strength in mathematics content and problem-solving likely affected their abilities to learn and apply math content in the programming context.

3. High socioeconomic status (SES) of participants. The high SES status of participants made them less transient and more consistent in attendance than middle grade students of lower SES status. There was no morbidity in the study and students were absent infrequently. This may have allowed for greater instructional consistency than is typically seen in the middle grades classroom.

209 Additionally, because the students were of high SES, they may have exhibited other differences from students who were not of high SES. For example, their access to video games and digital toys, as well as their previous experience with these entertainment devices, may have been differed when compared with other preteens and teens. Or their initial disposition towards mathematics may have differed in a way (see Limitation #4) that could have affected their engagement with the content addressed during video game and toy design and construction.

4. Favorable attitudes toward mathematics among participants. Because study participants possessed high attitude scores on all four scales of the ATMI, they were likely not representative of typical student populations. Because positive attitudes toward math correlate with high ability in math (Ma & Xu, 2004; LSAY, 2007; Yara, 2009) this factor may have affected the abilities of the study participants to learn and apply math content in the programming context.

5. Non-random assignment of groups. Study participants were not randomly assigned to the treatment and comparison groups. Treatment group students were honors math students who chose to enroll in math enrichment knowing that they would be engaged in research addressing the design and construction of video games and digital toys. Comparison group students were honors math student not simultaneously enrolled in math enrichment.

6. Differences in number of math courses in which students are enrolled.

Treatment group students were enrolled in two courses, consisting of their regularly scheduled math course and the elective math enrichment course in which the current research study was conducted. In contrast, the comparison group students were enrolled

210 only in their regularly scheduled math course. While treatment group students took more math than their comparison counterparts, the pre-to-post increase in scores on the studywide math content test were statistically similar for both groups.

7. Initial differences between groups. On achievement test scores, initial differences existed between treatment and comparison groups. However, no statistically significant differences existed between initial scores on studywide mathematics pretests.

Thus, although treatment group students may have possessed slightly greater overall mathematics content knowledge than comparison students, knowledge of mathematics content that was relevant to the research projects was comparable between groups. Initial differences also existed between groups with regard to ATMI scores. However, pre-to- post administrations of the ATMI revealed no statistically significant differences within groups. Had statistically significant differences existed within a group – indicating a significant increase or decrease in attitudes occurred during the study – then additional examination of those attitude scores and reasons for those changes would have required further investigation.

8. Researcher-created content tests not established as valid and reliable. Because the studywide and checkpoint mathematics tests had not been validated, they may not have tested what they intended to test. Validation of the questions would need to be accomplished via expert examination and revision. Further, additional administrations of the content tests would be required in order to establish test reliability.

9. Use of a single rater for representation ratings. Representation ratings were conducted by the researcher only. To improve ratings reliability, multiple raters could be employed and a measure of inter-rater reliability could be established.

211 Recommendations for Future Research

Results and analyses of the current research raised additional questions that should be examined in future research. The following recommendations suggest research that may resolve some of these questions.

1. Examination of replicability of results with different student populations.

Because the study participants in the current research possessed high math ability, high socioeconomic status, and favorable attitudes toward mathematics, they may not have been representative of typical student populations. Repeating the treatment interventions with different student populations may help reveal whether findings are generalizable to broader, and different, student populations.

2. Examination of concept mastery and transfer of concepts with different video game and digital toy projects as well as different programming environments. The selection of programming projects (e.g., Etch-a-Sketch, Frogger, and Tamagotchi Virtual

Pet) was predicated on researcher experience with instructing similar content in pilot research. However, other programming projects addressing similar or different NCTM content standards could have been developed and utilized with treatment group students.

Further, alternative programming environments (i.e., Scratch, Kodu, GameMaker) could have been employed as a vehicle for examining mastery of mathematics concepts and transfer to the traditional context.

3. Examination of the role of explicit bridging between contexts. The researcher avoided explicit bridging in this study. For example, there was no attempt to connect the

MicroWorlds EX programming code describing a line with the traditional, algebraic equation of a line. Without explicit bridging by the teacher between instructional contexts,

212 transfer may not be extensive. This may be especially true for contexts which are extremely different. Additionally, bridging may also be necessary for achieving more permanent transfer of concepts with students. While students may be able to deduce similarities between different contexts in the short term, these understanding may fade without bridging to clarify meaning. Thus, the role of bridging between the programming and the traditional context should be further examined.

4. Examination of the degree of correlation associated with transfer from various curricular models and instructional methods in mathematics. The current research found that design and construction of video games and digital toys via MicroWorlds EX projects was a viable method of mathematics instruction. However, study participants did not participate exclusively in one type of mathematics intervention. All participants took a traditional math course (typically honors math) while treatment group students also participated in elective math enrichment in which this research was conducted. Thus, the mastery of concepts demonstrated by treatment students in designing and constructing their game and toy projects can be solely attribute to the research intervention. The question still remains as to whether instruction of mathematics via video game design and construction differs in efficacy than other instructional methods. Specific methods – with groups taught math via a single method – should be studied. Further, specific programming environments should be examined to determine whether differences exist in their effectiveness in conveying mathematics through programming.

213 APPENDIX A

EXAMPLE TREATMENT STUDENT PROFILES

214 Example Treatment Student Profiles

The treatment group consisted of 5 female and 14 male preteenage and young teenage students enrolled in math enrichment. Students selected their own pseudonyms to use while working on research activities associated with this study. Following are two programmer profiles of students who engaged in video game design and construction and whose work is featured in this study.

Kandi

Kandi is a 6th-grade young woman enrolled in honors 6th grade mathematics. This is my third year teaching Kandi in math enrichment. She is the elder of two daughters; her younger sister is in fifth grade and looks like her twin. Kandi is calm and reserved and never seems worried about her work. She is warm and successful in making new friends, but she also possesses a quirky, dry sense of humor that she shares only with those whom she knows well. Kandi enters numerous academic competitions and has won several awards for writing, engineering and science. She is happy to win, but does not brag about her achievements. Kandi usually sits with Katt, another 6th-grade girl in the computer lab, but is comfortable working with anyone, male or female, or independently. In addition to being a good student, she is also a strong athlete and an avid Facebook fan. Her parents are involved in the school and her academics in a positive, supportive fashion.

During the video game study, Kandi was a minimalist in documenting her successes and challenges. She displayed moderate skill in identifying events and writing representations. She was quiet in class, but was sufficiently forthright to ask for help when she reached an impasse. Kandi seemed to appreciate the way MicroWorlds EX provided feedback, thereby allowing her to try out ideas and troubleshoot her own code.

215 Kandi enjoyed creating her pseudonym, and extended the “candy” theme into her projects. She seemed to like the fact that she could add color and artistic pizzazz to her games, which often showcased candy. Her Etch-a-Sketch featured a piece of candy as the drawing tool tip, and her Frogger involved helping a candy escape girls giving chase in a candy store. However, her Tamagotchi Virtual Pet, “Tinki,” was provided fruit (not candy) to eat and playtime activities (i.e., swimming, climbing) to enjoy.

Kandi invested extensive time and attention in developing her game graphics and programming code, going far beyond the basic video game models. She adapted and applied ideas from previous games to later games. For example, her Tinki Tamagotchi included motion commands in which Tinki traipsed across a river and back. This required

Kandi to painstakingly locate and incorporate the coordinates of each stepping-stone into her “skipping” procedure. She also included a drop-down list and added associated procedures, which she retrieved from a drawing project she completed in 5th grade, to allow the player to choose Tinki’s snack. Her finished Tinki Tamagotchi was one of the most complex games created during the course of the study, and her rich aesthetic and mathematical talents seemed well matched with the video game making task.

Richard

Richard Timeworth is the self-made moniker of a 6th grade young man currently taking prealgebra. This is my third year teaching Richard in math enrichment. He is the younger of two sons; his elder brother Dayne was also in the treatment group. The brothers got along reasonably well, but generally did not interact in class.

Richard is a small guy with an enormous spirit and a strong work ethic. He has artistic, scientific and strong mathematical talents that allowed him to make inventive,

216 attractive video games. Richard was always on-task and often volunteered an answer during group discussions. His strong personality – leaning towards the frenetic – caused him to feel more emotional about the work than his peers, but his investment and enthusiasm were impressive. He argued with me (humorously, knowing that I wouldn’t budge) about including “bullets, blood, and death” in his Frogger game when he knew I would allow only “paintballs, green goo, and stun.” His completed game showed a forest setting and a logger (Frogger) attempting to traverse it without being attacked by trees that moved with random, Brownian-style motion. For his Tamagotchi video game project,

Richard designed an aesthetically appealing World War II battlefield with soldiers and unusual variables including “Rally Troops.”

Richard was always vocal about discussing and troubleshooting his game. He was highly vested in writing events and representations, demonstrating a high level of performance on these tasks, and expressing frustration when he encountered difficulty.

Richard enjoyed sharing his work with me, the class and visiting guests. He was also an effective ambassador for video game programming among future generations. During a buddy activity with early childhood students, Richard requested to work with five-year- old Carson (my own son). Carson possesses basic experience in using MicroWorlds EX and the partnership produced interesting drawing featuring angled, colored lines all over the screen with the turtle working in Animate mode. They were all smiles. After class,

Richard said, “I couldn’t remember how to set pen width, but Carson knew how, so he reminded me!”

217

APPENDIX B

STUDYWIDE MATHEMATICS CONTENT TEST

218 STUDYWIDE MATHEMATICS CONTENT

PRE/POST Test (Circle One)

Student Pseudonym______Grade Level: ______

Directions: Solve each problem. Select and circle your answer from the choices given below the problem. There is only one best answer for each problem. You have 30 minutes.

Studywide Problem 1: The starting position of a dog is (-220, 0). On each step, the dog moves 30 units. He can walk North, East, South or West. If the dog steps North twice and East twice, what are the coordinates of his ending position?

N a) (0, 0) b) (-220, 60) Dog starting c) (-160, 60) position W E d) (60, 60) (‐220, 0) e) I don’t know

Studywide Problem 2: Which of the following sequences returns a jogger to her starting position?

a) Jog East 2 miles; jog North 2 miles b) Jog East 2 miles; jog East 2 miles c) Jog East 2 miles; jog South 2 miles d) Jog East 2 miles; jog West 2 miles e) I don’t know

219 Studywide Problem 3: An “undo” button on a calculator performs the inverse of the last operation. What operation “undoes” the operation + 8 ?

a) + 0 b) x 0 c) - 8 d) / 8 e) I don’t know

Studywide Problem 4: The North position on a circle is designated as 0 degrees. How many degrees of rotation around the circle coincides with the North position?

a) 180 degrees b) any integer multiple of 180 degrees c) 360 degrees d) any integer multiple of 360 degrees e) I don’t know

Studywide Problem 5: A car is travelling a straight-line path along y = x. At the origin (0, 0), the car makes a 180 degree turn. Along what line is it now travelling? N

a) y = x y = x b) y = -x (0, 0) c) y = 1/x W E d) y = -1/x e) I don’t know Car starting position

220 Studywide Problem 6: A car is travelling a straight-line path along y = x. At the origin (0, 0), the car makes a 90 degree turn. Along what line is it now travelling?

N a) y = x b) y = -x y = x c) y = 1/x (0, 0) d) y = -1/x W E e) I don’t know

Car starting position

Studywide Problem 7: Which of the following graphs shows x > 10?

a) ‐10 0 10 b) ‐10 0 10 c) ‐10 0 10 d) ‐10 0 10 e) I don’t know

Studywide Problem 8: Which equation describes the shaded region? N a) x > 50 (50, 75) b) x ≥ 50 c) y > 75 W E d) y ≥ 75 e) I don’t know

221 Studywide Problem 9: Assume that the shaded region is symmetrical. Which equations describe the shaded region? N a) -75 < x < 150 and -75 < y < 150 b) -150 < x < 150 and -75 < y < 75 c) -75 x 150 and -75 y 150 ≤ ≤ ≤ ≤ W E d) -150 ≤ x ≤ 150 and -75 ≤ y ≤ 75 e) I don’t know (150, ‐75)

Studywide Problem 10: A ladybug starts at the origin (0, 0). He crawls to the coordinates (-4, 3). What is the straight-line distance between his starting position and his new position? N a) 3 units b) 4 units c) 5 units (0, 0) W E d) 7 units Ladybug starting position e) I don’t know

Studywide Problem 11: Speed is equal to distance per time. On a computer screen, an animated car travels at 2 pixels per tenth of a second. Which of the following expressions represents a doubling of the car’s speed?

a) 1 pixel per tenth of a second b) 4 pixels per tenth of a second c) 1 pixel per second d) 2 pixels per second e) I don’t know

222 Studywide Problem 12: Highway vehicles move in lanes of traffic that run parallel to each other. Parallel lines have the same:

a) x-intercept b) y-intercept c) slope (m) d) coordinates e) I don’t know

Studywide Problem 13: Two objects intersect or “collide” if they share any points. Consider two, same-sized circles intersecting at a single point in a plane. If the area of each circle is 78.5 square units, what is the distance between the centers of the circles? (Note: Use 3.14 to approximate pi.)

a) about 5 units b) about 10 units c) about 12.5 units d) about 25 units e) I don’t know

Studywide Problem 14: A special clock measures time by ticking every tenth of a second. You want to track an event which occurs at one-minute intervals. For each one-minute interval, how many ticks must you count on the special clock?

a) 6 b) 60 c) 600 d) 6000 e) I don’t know

223 Studywide Problem 15: A runner executes one of his daily workouts according to the graph below. What could have happened between minutes 14 and 18?

a) The runner slowed down b) The runner sped up Miles c) The runner ran backwards d) The runner stopped running e) I don’t know 0 8 14 18 30 Time (minutes)

Studywide Problem 16: A variable, tennisshoe, gives the number of athletic shoes made at a factory. Tennisshoe equals 1500 after 1 hour; 4500 after 3 hours; and 9000 after 6 hours. What is the predicted value of tennisshoe after 11 hours? a) 9,000 b) 11,000 c) 15,000 d) 16,500 e) I don’t know

Studywide Problem 17: During the evening, the outside air cools down. The value of a variable, airtemp, decreases by 4 degrees Fahrenheit with the passage of 1 hour of time. After one hour, the new value is represented as:

airtemp - 4

During the morning, the outside air warms. The value of airtemp increases by 5 degrees Fahrenheit with the passage of one hour. The expression that gives the new value of the variable after one hour is… a) airtemp - 4 b) airtemp + 4 c) airtemp - 5 d) airtemp + 5 e) I don’t know

224 Studywide Problem 18: The variable lollipops gives the number of lollipops remaining in the classroom. The lollipops are eaten at the rate of 1 per 5 minutes until they are all gone. Which conditional statement correctly describes the incremental consumption of lollipops? a) decrease the number of lollipops by 1 b) if lollipops > 0 then decrease the number of lollipops by 1 c) if lollipops < 0 then decrease the number of lollipops by 1 d) if lollipops > 1 then decrease the number of lollipops by 1 e) I don’t know

Studywide Problem 19: The variable kenoball is a random number generated between 1 and 80, inclusive. What is the probability that kenoball is a single- digit number? a) 9/80 b) 10/80 c) 9/100 d) 10/100 e) I don’t know

Studywide Problem 20: A restaurant supply company ships several thousand fortune cookies to a take-out diner. Each fortune cookie contains one of 7 randomly-selected fortunes. Three fortunes are optimistic (“Today is your lucky day!”); three fortunes are neutral (“Brush your teeth twice daily”); and one fortune is pessimistic (“Your stocks will nosedive today”). If you select and read three fortune cookies, what is the probability that all the fortunes are positive? a) 3/7 b) 3 x 3/7 c) 3/7 + 3/7 + 3/7 d) 3/7 x 3/7 x 3/7 e) I don’t know

225

APPENDIX C

ATTITUDES TOWARD MATHEMATICS INVENTORY (ATMI)

226 227

228

APPENDIX D

CHECKPOINT MATHEMATICS CONTENT TESTS

229 ETCH-A-SKETCH

PRE/POST Test (Circle One)

Student Pseudonym______Grade Level: ______

Directions: Solve each problem. Select and circle your answer from the choices given below the problem. There is only one best answer for each problem. You have 15 minutes.

Etch Problem 1: The starting position of a ladybug is (30, 50). On each step, the ladybug moves 20 units. She can walk North, East, South or West. If the ladybug steps East once, South twice and West three times, what are the coordinates of her ending position? N a) (0, 0) Ladybug starting b) (-10, -10) position c) (-10, 10) (30, 50) W E d) (10, 10) e) I don’t know

Etch Problem 2: Which of the following sequences returns a bike to its starting position?

a) Ride East 2 miles; ride North 2 miles b) Ride West 2 miles; ride West 2 miles c) Ride East 2 miles; ride South 2 miles d) Ride North 2 miles; ride South 2 miles e) I don’t know

230 Etch Problem 3: An “undo” button on a calculator performs the inverse of the last operation. What operation “undoes” the operation + 5 ?

a) + 5 b) - 5 c) x 0 d) / 5 e) I don’t know

Etch Problem 4: The West position on a circle is designated as 270 degrees. How many degrees of rotation around the circle coincides with the West position?

a) 90 degrees b) any integer multiple of 90 degrees c) 360 degrees W d) any integer multiple of 360 degrees e) I don’t know

Etch Problem 5: Which of the following graphs shows x > 12?

a) ‐12 0 12 b) ‐12 0 12 c) ‐12 0 12 d) ‐12 0 12 e) I don’t know

231

Etch Problem 6: Which equation describes the shaded region?

N a) y > 10 (10, 20) b) y ≥ 20 c) x > 10 W E d) x ≥ 20 e) I don’t know

Etch Problem 7: Assume that the shaded region is symmetrical. Which equations describe the shaded region?

N a) -80 < x < 80 and -60 < y < 60 b) -80 < x < 60 and -80 < y < 60 c) -80 ≤ x ≤ 80 and -60 ≤ y ≤ 60 W E d) -80 ≤ x ≤ 60 and -80 ≤ y ≤ 60 e) I don’t know (80, ‐60)

232 FROGGER

PRE/POST Test (Circle One)

Student Pseudonym______Grade Level: ______

Directions: Solve each problem. Select and circle your answer from the choices given below the problem. There is only one best answer for each problem. You have 15 minutes.

Frogger Problem 1: A car is traveling a straight-line path with slope 2. The car makes a 180 degree turn. What is the slope of the line along which the car now travels? a) -2 b) 0 c) 2 d) undefined e) I don’t know

Frogger Problem 2: A car is travelling a straight-line path along y = x + 1. At (0, 1), the car makes a 180 degree turn. Along what line is it now travelling? N

a) y = x + 1 y = x + 1 b) y = -x - 1 (0,1) c) y = 1/x + 1 W E d) y = -1/x - 1 e) I don’t know

233

Frogger Problem 3: A car is traveling a straight-line path with slope 2. The car makes a 90 degree turn. What is the slope of the line along which the car now travels? a) -2 b) -1/2 c) 1/2 d) 2 e) I don’t know

Frogger Problem 4: A car is travelling a straight-line path along y = x + 1. At (0, 1), the car makes a 90 degree turn. Along what line is it now travelling? N

ITEM y = x + 1 (0,1) W E REMOVED Follows this line after turn

Frogger Problem 5: Speed is equal to distance per time. On a computer screen, an animated car travels at 7 pixels per tenth of a second. Which of the following expressions represents a doubling of the car’s speed? a) 7 pixels per tenth of a second b) 14 pixels per tenth of a second c) 7 pixels per second d) 14 pixels per second e) I don’t know

234 Frogger Problem 6: Cars move in lanes of traffic which run parallel to each other. Parallel lines have the same: a) x-intercept b) y-intercept c) slope (m) d) coordinates e) I don’t know

Frogger Problem 7: Perpendicular lines have slopes which: a) add up to 0 b) are equal c) multiply to 0 d) are negative reciprocals (multiply to -1) e) I don’t know

Frogger Problem 8: Two objects intersect or “collide” if they share any points. Consider two, same-sized circles intersecting at a single point in a plane. If the area of each circle is 314 square units, what is the distance between the centers of the circles? (Note: Use 3.14 to approximate pi.)

a) about 10 units b) about 20 units c) about 40 units d) about 100 units e) I don’t know

235

Frogger Problem 9: Two objects intersect or “collide” if they share any points. Consider two circles as shown. The radius of the big circle is 15 pixels. The radius of the small circle is 10 pixels. By what distance must the centers of the circles be separated to ensure the circles don’t collide? a) more than 5 pixels b) more than 10 pixels c) more than 25 pixels d) more than 30 pixels e) I don’t know

Frogger Problem 10: A special clock measures time by ticking every tenth of a second. You want to track an event which occurs at one-second intervals. For each one-second interval, how many ticks must you count on the special clock? a) 1 b) 10 c) 100 d) 1000 e) I don’t know

Frogger Problem 11: A runner executes one of his daily workouts according to the graph below. Compared with minutes 0 through 8, what happened when the runner entered the interval of minutes 8 through 14?

a) The runner slowed down b) The runner sped up c) The runner ran backwards Miles d) The runner stopped running e) I don’t know 0 8 14 18 30 Time (minutes)

236 TAMAGOTCHI VIRTUAL PET

PRE/POST Test (Circle One)

Student Pseudonym______Grade Level: ______

Directions: Solve each problem. Select and circle your answer from the choices given below the problem. There is only one best answer for each problem. You have 15 minutes.

Tamagotchi Problem 1: A variable, basketball, gives the number of balls made at a factory. Basketball equals 2000 after 1 hour; 6000 after 3 hours; and 12000 after 6 hours. What is the predicted value of basketball after 9 hours? a) 9,000 b) 14,000 c) 18,000 d) 21,000 e) I don’t know

Tamagotchi Problem 2: A variable, piggybank gives the amount of money in a child’s bank. The initial value of piggybank is $5. Piggybank increases by $6 every week. so that at the end of Week 1, the child has $11 in the bank. What is the value of piggybank at the end of Week 8 (assuming no withdrawals are made)? a) $48 b) $53 c) $59 d) $60 e) I don’t know

237 Tamagotchi Problem 3: During the current economic depression, membership at a country club decreases each week. The value of a variable, members, decreases by 3 people with the passage of each week. After one week, the new value is represented as:

members - 3

Management lowers membership fees to create an incentive for people to join the club. Now, people no longer leave, and the value of members increases by 2 people with the passage of each week. The expression which gives the new value of the variable after one week is… a) members - 2 b) members + 2 c) members - 3 d) members + 3 e) I don’t know

Tamagotchi Problem 4: The variable pancakes gives the number of pancakes remaining at a pancake breakfast. The pancakes are eaten at the rate of 1 per 4 minutes until they are all gone. Which conditional statement correctly describes the incremental consumption of pancakes?

a) decrease the number of pancakes by 1 b) if pancakes > 0 then decrease the number of pancakes by 1 c) if pancakes < 0 then decrease the number of pancakes by 1 d) if pancakes > 1 then decrease the number of pancakes by 1 e) I don’t know

238 Tamagotchi Problem 5: The variable dicethrow is a random number generated between 1 and 6, inclusive. What is the probability that dicethrow is greater than 4? a) 0 b) 1/6 c) 2/6 d) 3/6 e) I don’t know

Tamagotchi Problem 6: The value of the variable happiness ranges from 0 to 100 and changes in increments of 5. When happiness = 0, a “dead” message is shown. When happiness < 20, a “sad” message is shown. When happiness > 85, an ”overjoyed” message is shown. No message is shown when happiness possesses other values. When happiness = 20, what message is shown? a) a “dead” message b) a “sad” message c) an “overjoyed” message d) no message e) I don’t know

Tamagotchi Problem 7: A restaurant supply company ships several thousand fortune cookies to a take-out diner. Each fortune cookie contains one of 10 randomly-selected fortunes. Four fortunes are optimistic (“Today is your lucky day!”); four fortunes are neutral (“Brush your teeth twice daily”); and two fortunes are pessimistic (“Your stocks will nosedive today”). If you select and read three fortune cookies, what is the probability that all the fortunes are positive? a) 4/10 b) 3 x 4/10 c) 4/10 + 4/10 + 4/10 d) 4/10 x 4/10 x 4/10 e) I don’t know

239 APPENDIX E

EVENTS LIST TEMPLATE

240 Name ______Gr ____

MODEL VIDEO GAME EVENT ANALYSIS

List as many events as you can identify to fully describe ______:

#10

#11

#12

#13

If needed, continue to next page…

241 APPENDIX F

REPRESENTATIONS TEMPLATE (ETCH-A-SKETCH)

242

243

244 APPENDIX G

REFLECTIONS AND PLANS DESIGN JOURNAL TEMPLATES

245

246

247 APPENDIX H

TRANSCRIPT OF CLASS DISCUSSION (ETCH-A-SKETCH)

248 Treatment Class Discussion Constructing the Etch-a-Sketch December 3, 2008 Transcript

Trans03Dec2008Part1 0:00 [Background noises of students mimicking Bob voice from Monsters vs. Aliens] RICHARD Mrs. McCue! RSCHR Yes sir. RICHARD It won’t work, it’s too big. RSCHR OK. Um, we can either try to just erase it, or – let’s go back. Open up my website and retrieve that same picture again. RICHARD OK. RSCHR ‘Cause then we’ll just slap it right over the top and cover it up. RICHARD [working] RSCHR [To class] Please help each other and talk about it…

1:01 RSCHR [To Katt] Yes ma’am? KATT I hit the clean button and it took away my frame… RSCHR Oh, so what happened, how do you think that happened? You forgot to do what? KATT I forgot to freeze it but I unfreezed it ‘cause I was gonna draw over it. I messed up my duck. RSCHR Do you want to go with the red frame again? KATT No, no. RSCHR You can draw it. So here, take your drawing tools. KATT Do I just free draw it? ‘Cause then it’s gonna be all sloppy. RSCHR Well no, because you can use it as a square. So look, let’s say you fill in the back and you make it pink. KATT Mmm, hmm. RSCHR And then you can use this square and make the inside, uh, yellow. Or blue. Or this kind of aqua? KATT Aqua! RSCHR OK, so take that, and just draw a square. KATT Cool! RSCHR Isn’t that neat? KATT Can I go back and round the edges? RSCHR Um, I don’t know…. can you do that?... KATT Like if I take a pencil and… RSCHR Yes, yes! Somehow you could that, so you can adjust it… How’s it going?

249 1:48 NICK How do you make that? RSCHR Oh, this was right except you needed the space. So, hold down – put your finger on the control key. Now, click on that. And then use the dots and stretch it in. There you go… and then you can make it however small you want, OK. Alright, how’s everybody doing? UKNOWN Good. RSCHR Let me, uh… when you save, remember I need you to save it with a unique name meaning you need to change the name. Yeah, change the date. So don’t save it over your old file, save it as – what’s today? – save it as NameDec3Etch. Perfect!

2:56 PHIL OK, I need diagonally… RSCHR Ah, yes, sneaky…. You know how to make the buttons, Alexander? ALEX Nooo. RSCHR OK, go to the finger… ALEX The finger? MATITIO [in background] The FINGER… RSCHR Uh-huh. ALEX And click here? RSCHR Anywhere you want, you can always move it. Just click and it’s going to open up a box. OK, now, um, the Label would be like, left. ALEX OK PHIL [in background] Oh, let’s see uh… RYAN [in background] What’s 40… uh, 125 PHIL it’s 90…., uh, 135, 135, exactly… RSCHR And then the instruction – what do you think the instruction is? First Pen Down, right, PD; and then space, and then let’s turn him to point the right direction – the boy’s name set heading, SETH, S- E-T-H; space. ALEX Errrr… RSCHR Uh-huh, that’s right. And then remember, it starts at 0 at the top. And goes to 90, 180, 270. So left is 270, right. ALEX OK. RSCHR So that just points him, but now we want him to go forward. So FD space 5. ALEX [nodding] RSCHR So now you can try it. ALEX [clicking button and seeing that it works] RSCHR You see it worked. ALEX OK. RSCHR Perfect!

250 4:16 RSCHR If you want to adjust the size of the button, what you need to do is hold down the control key and then click on the button and it will give you four sizing dots, and you can adjust the size. RYAN [in background] Shake, shake, shake shake… RSCHR If you make it too small… UK [inaud] RSCHR Right, so that’s where you may want to use an abbreviation. RICHARD Mrs. McCue… one instruction PD… [inaud] RSCHR OK, so hold down the control key… click on it… stretch it out. RICHARD Oh, I didn’t know I could do that. RSCHR Yeah. RICHARD And it won’t work, though, it won’t let me do the PD. RSCHR Which one? [McCue looking] RICHARD See the Commence Droppings. So then not to do it anymore. RSCHR Oh, well look at what the command is. What is the command on Commence Droppings? RICHARD PD RSCHR OK, so PD means what? RICHARD Pen Down RSCHR Right. RICHARD Oh! How do I do pen up? RSCHR Make a new button and that’ll be the command. RICHARD What’s it, P-U? RSCHR PU. RICHARD Oh! I get it. RSCHR [chuckles]

5:10 KANDI I did everything and it goes diagonally down and diagonally up. RSCHR You want it to go straight up? KANDI Up, straight up. RSCHR That’s zero. KANDI Oh… RYAN When I finish this, I’m going to bring my flash drive to school so I can keep it. RSCHR That’ll be awesome. PHIL [to neighbor] Wait! What did I do on the down left that’s messed, that’s not… oh, PD SETH – it’s not doing anything. RSCHR Good – so you did what? PHIL If forgot the FD part… RSCHR Right, you pointed him, you just didn’t make him go forward.

251 5:46 KATT Wait, um, it’s freezing up. It’s being like weird. Look, I clicked on that ‘cause I want to put that in the background. Look it won’t do that so it’s being weird. RSCHR It is being weird. Let’s do this…. Save Project As… OK, we’re gonna change the date…. Close it and just open it back up. Now go…

6:15 ? Look at mine. RSCHR Cooool… he’s a penguin in a cage. JOE Can I change this, the penpoint… RT [Interrupting In background] Mrs. McCue, there’s no… RSCHR [To Joe] Yeah, you can change it to make it look like anything you want. So go to Shapes…. JOE Since it doesn’t look good, since there’s like lines in the middle. DAYNE [Interrupting In background] Mrs. McCue, look what I did… RSCHR Yeah. Double-click on a shape spot. And now make anything new that you want and then apply it to the turtle. DAYNE Look what I did. I combined this picture with this picture. RSCHR Very nice. So he looks like he’s in a cage? Is that the idea, or a box. DAYNE Like I took this one, but then I changed the colors and I put the penguin there so. RSCHR OK, cool. DAYNE So, I don’t know. I guess he’s like looking through a mirror or something. RSCHR He’s uh, in a house – looking through a grate. JONIE Mrs. McCue, can you change the color of your buttons? RSCHR Good question. No, unfortunately MicroWorlds is limited. That would be cool. JONIE Oh… RSCHR The only thing you can do is you can make a turtle look like a button and draw like a box and make anything you want, but that’s a little more complicated. But yeah, unfortunately you can’t. And you can’t change the text font or anything like that.

7:20 RSCHR Alright, I’m going to give you a couple more minutes, then we’re going to talk about the constraints so the drawing tip doesn’t go off into the frame. Yes, Matitio. MATITIO Um, how do you freeze it. When I clicked Clean it all goes away. I had the whole thing and then I clicked Clean and… RSCHR Right. Any lines that are drawn will go away. MATITIO How do you freeze it? RSCHR So right-click on here…. Um… it’s all frozen.

25 2 MATITIO Oh, so you can’t write anything? RSCHR No, that part, if you want to write ET on there, then you have to freeze again. UNKNOWN What’s ET? MATITIO How do you freeze it then. RSCHR So unfreeze and then refreeze it. Unfreeze it, draw what you want and then refreeze it. MATITIO Oh, all over. RSCHR Sorry.

8:00 [Background noise] RB [In background] You actually want to go all the way around… DRAKE 270 plus 45 is 315! RSCHR There you go! RB Sounds like a good one, yeah. [Background noise] NICK I’m almost done, now all I need is South – no – North, uh… RSCHR Which one are you missing? NICK I’m missing Northwest. RSCHR OK, now. Tell me why you decided to orient the buttons around the Clean like that. NICK Because it looks better. It looks more neat. RSCHR But how come Southwest is up there and how come Northeast is… NICK Oh, sorry. Yeah, I know. I was messing up a little bit, so… RSCHR No, I’m just curious. Are you choosing to put them in these positions for a reason? NICK No, not really. RSCHR No? Is Southwest more logical here or here. NICK [pointing correctly] Here, because West is going that way. RSCHR OK, so you’re using it to represent visually where those directions are. NICK Yeah. RSCHR OK.

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Trans03Dec2008Part2 0:00 KATT She figured out how to make it curve! RSCHR How did you do that? CHLOE Um, you program a circle and… RSCHR [Gasps] How smart! KATT Awesome! RSCHR You are awesome, that’s cool… taught me something new.

253 NICK What happened? KATT Um can I just put RD? RSCHR For what Right Down? Yeah. KATT Chloe You can make zigzags. NICK [to Chloe] What did you do, you made it curve? RSCHR Yeah! CHLOE You click on the circle. NICK What happens? CHLOSE You make a rounded square. NICK Ohhhh…. A rounded square. 0:36

1:45 RICHARD Look look look. Activate Droppings and Deactivate Droppings. RSCHR I like that. RICHARD I don’t know where to put the North button ‘cause it won’t fit.

2:14 RSCHR [in response to question] How do you jump? You could make it forward a bigger amount. RICHARD Oh – no – I know how to jump. You just go forward without putting pen down! DAYNE Oh yeah, that’s how you do it! RSCHR I like that. That was smart. DRAKE I make a square! RSCHR You make a square, awesome! You mean using your etch-a- sketch? DRAKE Yeah. And, what do you want us to do with the save? RSCHR OK, everybody when you save it needs to be your first name, or some abbreviation of that and then December 3 then Etch.

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Trans03Dec2008Pt3 0:00 [In response to question] RSCHR Um , the only way you can do that, you can make a button called Pen Erase. PHIL OK. RSCHR You haven’t learned Pen Erase. Pen Erase will go back and erase the line you just drew. Buut…. what direction do we need to go? PHIL Well, I have all eight of the directions. RSCHR So, it depends on what direction he came from, alright? That’s hard to know. How about this… Let’s say you set me to this direction and I went forward and drew the line. And now you go,

254 ‘Shoot, I want to undo that.’ What do I need to do to erase that. Go BACK that same direction, right? PHIL But it’s just gonna do the line, though. RSCHR No, if you put in PE for pen erase, space, BK 5 – I go into erase mode and back up that distance. NICK/RYAN Oh, yeah, that’s it. PHIL Thank you Mrs. McCue. RSCHR Sure. And he’s already facing whatever direction you want him to be faced. The only questions I don’t know is, I don’t know whether it will try to erase the gray under it. PHIL No, no because the background’s locked (frozen). RSCHR That’s good… That’s neat, I like that. RYAN Is this how you spell Erase? RSCHR Uh-huh. RYAN Now, P-what? RSCHR PE space BK 5. PHIL It worked. RSCHR So he’s gonna stay facing whatever direction he was and now he’s gonna erase the line. That’s neat. I like that you guys. 1:35

1:44 [Angelie working on her tool tip so that she can she where it is – she’s concerned about it being too small.]

1:55 RICHARD Look I found how to do it, Mrs. McCue. Look look look! I do PE, then Activate Droppings North, right. You set it to something, then you do Undo, and then you go over it and it disappears. RSCHR Cool! You could even maybe make it into the undo where you don’t have to go over it. RICHARD Well I like that. 2:13 CROWD [Lots of chatter regarding the details of erasing.] 2:22 RSCHR You don’t have to identify the direction, do you? NICK Oh yeah. RSCHR You could just say, he’s already facing the right direction, we just want him to go backwards. DAYNE Ugh, I pressed cleannnn…. RSCHR What did you want to do [Joseph]? JOE No, actually, I figured it out. DAYNE I pressed clean. RSCHR Uh-oh. You can go back to my website to get the frame… because what did you not do? DAYNE I stamped it. RSCHR You stamped it but you didn’t……

255 DAYNE What? RSCHR …freeze. DAYNE Oh, you have to freeze it? RSCHR [Laughs] Yeah. [To Randy] OK, I’m going to have them work on this next step. RB OK. RSCHR [To class] Everybody! I need your attention on this next part. Stop what you’re doing. Spin your chairs. I need to see all your faces that way – for the next three minutes. All chairs, spin your chairs – spin, spin, turn. OK. Look up at my screen. This is my drawing tool tip, right. Not nearly as beautiful as yours… I realize that. I’m going to move it to this leftmost barrier. Who knows how I can tell what the coordinates are? [Andrew raises hand.] Andrew? ANDREW Uh, you open its backpack. RSCHR Open its backpack. How do I open the backpack? ANDREW Right-click and Open Backpack. RSCHR Alright. Alright. Now, I’m at the state tab. What are the coordinates right here? JOE Negative 140 and… 8. RSCHR Which part of that is the x part? GROUP Negative 140. RSCHR So if I pick up this guy and move him more this way [horizontally]… RICHARD Oooo, you can do that? RSCHR Yeah, ‘cause he’s unfrozen right now. RICHARD Oh yeah yeah. RSCHR His x-coordinate becomes what? GROUP Negative 171. RSCHR Negative 171. It’s becoming more negative. Right? But what I care about is this boundary. Now as I move him, if I stay right at this boundary and I move him up and down… do you see that the x- coordinate is staying the same? GROUP [Quiet agreement] RSCHR What’s happening, though, to the y-coordinate. If I move up, the y- coordinate becomes more… GROUP Increases. RSCHR More positive or more negative? GROUP [Emphatically] More positive. RSCHR But if I go down, it becomes more… GROUP [Emphatically] Negative. RSCHR Negative. OK, but the x part is staying the same. It’s staying at negative 144. It’s probably the same for you. If you use my frame, it’s probably that if you stick it right at that barrier there, it’s probably negative 144.

256 4:40 RSCHR Here’s the challenging part. We’re gonna do it together for this leftmost barrier and then I want to see if you can extend it to the top, right and bottom. Alright, so if I know that -144 is that barrier, then I’m gonna go over to the Rules tab…. and I’ve actually gotten them written in right here so I’m look just at this one right here…. can you guys see that fairly well? I’m going to read it out, ‘cause it’s fairly small… it says, XCOR… space… less than… negative – and I’m got 142 – so maybe the first time I did this, I was over to the right a couple of pixels; BK… space 5. OK? So what does all that mean? GROUP [Chatter, many answering] RSCHR Angelie, let me start with you. ANGELIE Um, if you go past negative 140, or 142 pixels… RSCHR Right… ANGELIE um, it will – the rule is it has to go back five pixels. RSCHR Right. So it says, WHEN THIS – when I go more negative than 142 (or -144, whatever my number is) – then the consequence, DO THAT, is BK 5.

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257 APPENDIX I

TAMAGOTCHI VIDEO GAME MODEL PROGRAMMING CODE WITH GAPS

258

259

APPENDIX J

UNLV IRB APPROVALS

260 261 262 263 APPENDIX K

CONSENT FORMS

264

PARENT PERMISSION FORM

Department of Curriculum and Instruction

TITLE OF STUDY: Middle School Math Skills Development through Student Produced Video Games INVESTIGATOR(S): Randall Boone, Ph.D. and Camille McCue, MA CONTACT PHONE NUMBER: 702-895-3233 (Randall Boone) or 702-806-8052 (Camille McCue)

Purpose of the Study Your child is invited to participate in a research study. The purpose of this study is to measure the mathematics skills acquired by middle school students as they construct video games.

Participants – TREATMENT GROUP Your child is being asked to participate in the study as a member of the treatment group. All math enrichment students taking the “Math Enrichment” course instructed by at The Alexander Dawson School beginning Fall 2008 are potential members of the treatment group. Treatment group participants will analyze the mathematics of video games and construct their own video games. Other students who are not enrolled in math enrichment are being invited to serve as the comparison group. Comparison group students provide pre and post-treatment “standards” against which treatment group math performance and affective disposition can be compared.

Procedures If you allow your child to volunteer to participate in this study, work your child produces during the study and observations of your child at work will be included in the study results. Whether you and your child choose to participate in the study, your child will engage in the same learning opportunities as all members of the math enrichment course. Approximately one-half of the math enrichment course work will address learning activities related to the study. During class meetings related to the research study, your child will perform several tasks. Your child will be asked to take a short math pre-test at the start of the Fall semester and take a short math post-test at the end of the Fall semester to measure mathematical knowledge as relates to video game design. The test runs approximately 30 minutes in duration. Additionally, your child will engage in “treatment” activities, analyzing the mathematics of video games and then synthesizing and programming their own video games. The games are specifically chosen for the mathematical problem-solving concepts and tasks they entail. To obtain qualitative data

265 regarding your child’s progress, your child will be asked to maintain a design notebook in which he/she records game ideas, strategies and problem-solving activities. Computer files of games your child produces will also be saved and assessed regarding the application of mathematics in the video game context. Your child will also be interviewed by the teacher/researcher (Camille McCue) regarding his/her mathematical thinking throughout game analysis and construction. The interviews will be audio recorded. Lastly, your child will engage in an attitude survey (at the beginning of the semester and again at the end of the semester) to measure his/her disposition towards mathematics. This survey runs approximately 15 minutes in duration.

Benefits of Participation There may be direct benefits to your child as a participant in this study. It is probable that treatment group participants will acquire basic skills in analyzing video game design; performing computer programming; and solving standards-based mathematics problems. All participants may also experience benefits from self-reflection regarding personal and professional interest in mathematics and programming video games. With regard to the overall impact on society, information regarding impact of treatment activities on participants' cognitive skills in mathematics and affective disposition towards mathematics may be used to modify and improve curriculum and instruction in middle school mathematics instruction.

Risks of Participation There are risks involved in all research studies. This study includes only minimal risks. Social risk is minimal. Math pre and post-tests, video game analysis and construction activities, design notebooks, interviews and attitude surveys will have no bearing on classroom grades and will not be accessible to other school personnel besides Mrs. McCue. Student identification on documents will be coded (using pseudonyms) to obscure the connection between the actual student and the work produced.

Cost /Compensation There will not be financial cost to you to participate in this study. The study will require approximately 37 hours of your child’s time, during regularly scheduled math enrichment class hours. Non-participants will take part in the same math enrichment activities as study participants, but their work will not be included in the study results. Your child will not be compensated for their time.

Contact Information If you or your child have any questions or concerns about the study, you may contact Randall Boone at (702) 895-3233 or Camille McCue at (702) 806-8052. For questions regarding the rights of research subjects, any complaints or comments regarding the manner in which the study is being conducted you may contact the UNLV Office for the Protection of Research Subjects at 702-895-2794.

266 Voluntary Participation Your child’s participation in this study is voluntary. There is no pressure to participate as your child will engage in the same learning activities regardless of whether you and your child choose to participate in the study. Your child may refuse to participate in this study or in any part of this study. Your child may withdraw at any time without prejudice to your relations with the university. You or your child are encouraged to ask questions about this study at the beginning or any time during the research study.

Confidentiality All information gathered in this study will be kept completely confidential. No reference will be made in written or oral materials that could link your child to this study. All records will be stored in a locked facility at UNLV for 3 years after completion of the study. After the storage time, the original, paper documents will be shredded. Electronic data (including digital audio recordings) will be stored as a password-protected file on a password-protected system until completion of publications, then stored to removable media and placed in a secure file cabinet where it will be stored indefinitely.

Participant Consent: I have read the above information and agree for my child to participate in this study. I am at least 18 years of age. A copy of this form has been given to me.

Signature of Parent Child’s Name (Please print)

Parent Name (Please Print) Date

I agree to allow my child to be audio taped for the purpose of this research study.

Signature of Parent Child’s Name (Please print)

Parent Name (Please Print) Date

Participant Note: Please do not sign this document if the Approval Stamp is missing or is expired.

267

PARENT PERMISSION FORM

Department of Curriculum and Instruction

Participants – COMPARISON GROUP Your child is being asked to participate in the study as a member of the comparison group. Comparison group students are high-performing math students (6th and 7th grade students taking “Honors Math”) who are not enrolled in the math enrichment exploratory course (“Math Enrichment”). Math enrichment students comprise the treatment group; they will be programming video games beginning Fall 2008. Comparison group students provide pre and post-treatment “standards” against which treatment group math performance and affective disposition can be compared.

Procedures If you allow your child to volunteer to participate in this study, your child will be asked to take a short math pre-test and attitude survey at the start of the Fall semester and again at the end of the semester. The math test runs approximately 30 minutes in duration and the attitude survey runs approximately 15 minutes in duration.

Benefits of Participation There may not be direct benefits to your child as a participant in this study. However, it is probable that treatment group participants will acquire basic skills in analyzing video game design; performing computer programming; and solving standards-based mathematics problems. All participants may also experience benefits from self-reflection regarding personal and professional interest in mathematics and programming video games. With regard to the overall impact on society, information regarding impact of treatment activities on participants' cognitive skills in mathematics and affective

268 disposition towards mathematics may be used to modify and improve curriculum and instruction in middle school mathematics instruction.

Risks of Participation There are risks involved in all research studies. This study includes only minimal risks. Social risk is minimal. Math pre and post-tests and attitude surveys will have no bearing on classroom grades and will not be accessible to other school personnel besides Mrs. McCue. Student identification on documents will be coded (using pseudonyms).

Cost /Compensation There will not be financial cost to you to participate in this study. The study will take 1.5 hours of your child’s time, during the regularly scheduled Honors Math class period. Your child will not be penalized for time missed in the classroom during their participation in the study. Non-participants will engage in the same content tests and attitude surveys, but their results will not be used in the study. Your child will not be compensated for their time.

Voluntary Participation Your child’s participation in this study is voluntary. There is no pressure to participate as your child will engage in the same learning activities regardless of whether you and your child choose to participate in the study. Your child may refuse to participate in this study or in any part of this study. Your child may withdraw at any time without prejudice to your relations with the university. You or your child are encouraged to ask questions about this study at the beginning or any time during the research study.

Confidentiality All information gathered in this study will be kept completely confidential. No reference will be made in written or oral materials that could link your child to this study. All records will be stored in a locked facility at UNLV for 3 years after completion of the study. After the storage time, the original, paper documents will be shredded. Electronic data will be stored as a password-protected file on a password-protected system until completion of publications, then stored to removable media and placed in a secure file cabinet where it will be stored indefinitely.

Participant Consent: I have read the above information and agree to participate in this study. I am at least 18 years of age. A copy of this form has been given to me.

269

Signature of Parent Child’s Name (Please print)

Parent Name (Please Print) Date

Participant Note: Please do not sign this document if the Approval Stamp is missing or is expired

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289 VITA

Graduate College University of Nevada, Las Vegas

Camille Moody McCue

Degrees: Bachelor of Arts in Mathematics, 1988 University of Texas, Austin

Master of Arts in Curriculum & Instruction, 1993 University of Texas, San Antonio

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Dissertation Title: Learning Middle School Mathematics Through Student Designed and Constructed Video Games

Dissertation Examination Committee: Chairperson, Randall Boone, Ph. D. Committee Member, Kent Crippen, Ph.D. Committee Member, PG Schrader, Ph.D. Graduate Faculty Representative, David James, Ph.D., PE

291